MathDB

2007 F = Ma

Part of F = Ma

Subcontests

(38)
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2007 F = ma #36: Resultant Length of Rope

A point object of mass mm is connected to a cylinder of radius RR via a massless rope. At time t=0t = 0 the object is moving with an initial velocity v0v_0 perpendicular to the rope, the rope has a length L0L_0, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds TmaxT_{max}. Express your answers in terms of TmaxT_{max}, mm, L0L_0, RR, and v0v_0. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the length (not yet wound) of the rope?
<spanclass=′latex−bold′>(A)</span> L0−πR <span class='latex-bold'>(A)</span>\ L_0 - \pi R
<spanclass=′latex−bold′>(B)</span> L0−2πR <span class='latex-bold'>(B)</span>\ L_0 - 2 \pi R
<spanclass=′latex−bold′>(C)</span> L0−18πR <span class='latex-bold'>(C)</span>\ L_0 - \sqrt{18} \pi R
<spanclass=′latex−bold′>(D)</span> mv02Tmax <span class='latex-bold'>(D)</span>\ \frac{mv_0^2}{T_{max}}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #35: Instantaneous Kinetic Energy

A point object of mass mm is connected to a cylinder of radius RR via a massless rope. At time t=0t = 0 the object is moving with an initial velocity v0v_0 perpendicular to the rope, the rope has a length L0L_0, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds TmaxT_{max}. Express your answers in terms of TmaxT_{max}, mm, L0L_0, RR, and v0v_0. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the kinetic energy of the object at the instant that the rope breaks?
<spanclass=′latex−bold′>(A)</span> mv022 <span class='latex-bold'>(A)</span>\ \frac{mv_0^2}{2}
<spanclass=′latex−bold′>(B)</span> mv02R2L0 <span class='latex-bold'>(B)</span>\ \frac{mv_0^2R}{2L_0}
<spanclass=′latex−bold′>(C)</span> mv02R22L02 <span class='latex-bold'>(C)</span>\ \frac{mv_0^2R^2}{2L_0^2}
<spanclass=′latex−bold′>(D)</span> mv02L022R2 <span class='latex-bold'>(D)</span>\ \frac{mv_0^2L_0^2}{2R^2}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #34: Point object in a cylinder

A point object of mass mm is connected to a cylinder of radius RR via a massless rope. At time t=0t = 0 the object is moving with an initial velocity v0v_0 perpendicular to the rope, the rope has a length L0L_0, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds TmaxT_{max}. Express your answers in terms of TmaxT_{max}, mm, L0L_0, RR, and v0v_0. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy] What is the angular momentum of the object with respect to the axis of the cylinder at the instant that the rope breaks?
<spanclass=′latex−bold′>(A)</span> mv0R <span class='latex-bold'>(A)</span>\ mv_0R
<spanclass=′latex−bold′>(B)</span> m2v03Tmax <span class='latex-bold'>(B)</span>\ \frac{m^2v_0^3}{T_{max}}
<spanclass=′latex−bold′>(C)</span> mv0L0 <span class='latex-bold'>(C)</span>\ mv_0L_0
<spanclass=′latex−bold′>(D)</span> TmaxR2v0 <span class='latex-bold'>(D)</span>\ \frac{T_{max}R^2}{v_0}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #30: Maximum Skid Acceleration

A simplified model of a bicycle of mass MM has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is ww, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude aa. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("hh",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("ww",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 2 (Question 30): Assume, instead, that the coefficient of sliding friction between each tire and the ground is different: μ1\mu_1 for the front tire and μ2\mu_2 for the rear tire. Let μ1=2μ2\mu_1 = 2\mu_2.
Assume that both tires are skidding: sliding without rotating. What is the maximum value of aa so that both tires remain in contact with the ground?
<spanclass=′latex−bold′>(A)</span> wgh <span class='latex-bold'>(A)</span>\ \frac{wg}{h}
<spanclass=′latex−bold′>(B)</span> wg3h <span class='latex-bold'>(B)</span>\ \frac{wg}{3h}
<spanclass=′latex−bold′>(C)</span> 2wg3h <span class='latex-bold'>(C)</span>\ \frac{2wg}{3h}
<spanclass=′latex−bold′>(D)</span> hg2w <span class='latex-bold'>(D)</span>\ \frac{hg}{2w}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #29: Maximum Acceleration

A simplified model of a bicycle of mass MM has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is ww, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude aa. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("hh",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("ww",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 1 (Questions 28 - 29): Assume that the coefficient of sliding friction between each tire and the ground is μ\mu, and that both tires are skidding: sliding without rotating. Express your answers in terms of ww, hh, MM, and gg.
What is the maximum value of aa so that both tires remain in contact with the ground?
<spanclass=′latex−bold′>(A)</span> wgh <span class='latex-bold'>(A)</span>\ \frac{wg}{h}
<spanclass=′latex−bold′>(B)</span> wg2h <span class='latex-bold'>(B)</span>\ \frac{wg}{2h}
<spanclass=′latex−bold′>(C)</span> hg2w <span class='latex-bold'>(C)</span>\ \frac{hg}{2w}
<spanclass=′latex−bold′>(D)</span> h2wg <span class='latex-bold'>(D)</span>\ \frac{h}{2wg}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #28: Maximum Friction Coefficient

A simplified model of a bicycle of mass MM has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is ww, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude aa. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("hh",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("ww",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 1 (Questions 28 - 29): Assume that the coefficient of sliding friction between each tire and the ground is μ\mu, and that both tires are skidding: sliding without rotating. Express your answers in terms of ww, hh, MM, and gg.
What is the maximum value of μ\mu so that both tires remain in contact with the ground?
<spanclass=′latex−bold′>(A)</span> w2h <span class='latex-bold'>(A)</span>\ \frac{w}{2h}
<spanclass=′latex−bold′>(B)</span> h2w <span class='latex-bold'>(B)</span>\ \frac{h}{2w}
<spanclass=′latex−bold′>(C)</span> 2hw <span class='latex-bold'>(C)</span>\ \frac{2h}{w}
<spanclass=′latex−bold′>(D)</span> wh <span class='latex-bold'>(D)</span>\ \frac{w}{h}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #38: Ratio of Kinetic Energies

A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds TmaxT_{max}. One end of the cord is attached to a fixed point, the other is attached to an object of mass 3m3m. If a second, smaller object of mass m moving at an initial speed v0v_0 strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of vfv_f. All motion occurs on a horizontal, frictionless surface.
Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision.
<spanclass=′latex−bold′>(A)</span> 1/4<spanclass=′latex−bold′>(B)</span> 1/3<spanclass=′latex−bold′>(C)</span> 1/2<spanclass=′latex−bold′>(D)</span> 3/4<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(A)</span>\ 1/4 \qquad<span class='latex-bold'>(B)</span>\ 1/3 \qquad<span class='latex-bold'>(C)</span>\ 1/2 \qquad<span class='latex-bold'>(D)</span>\ 3/4 \qquad<span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #37: Elastic Cord Collision

A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds TmaxT_{max}. One end of the cord is attached to a fixed point, the other is attached to an object of mass 3m3m. If a second, smaller object of mass m moving at an initial speed v0v_0 strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of vfv_f. All motion occurs on a horizontal, frictionless surface.
Find vf/v0v_f/v_0.
<spanclass=′latex−bold′>(A)</span> 1/12<spanclass=′latex−bold′>(B)</span> 1/2<spanclass=′latex−bold′>(C)</span> 1/6<spanclass=′latex−bold′>(D)</span> 1/3<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(A)</span>\ 1/\sqrt{12}\qquad<span class='latex-bold'>(B)</span>\ 1/\sqrt{2}\qquad<span class='latex-bold'>(C)</span>\ 1/\sqrt{6} \qquad<span class='latex-bold'>(D)</span>\ 1/\sqrt{3}\qquad<span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #33: Maximum Value

A thin, uniform rod has mass mm and length LL. Let the acceleration due to gravity be gg. Let the rotational inertia of the rod about its center be md2md^2.
The rod is suspended from a distance kdkd from the center, and undergoes small oscillations with an angular frequency βgd\beta \sqrt{\frac{g}{d}}.
Find the maximum value of β\beta.
<spanclass=′latex−bold′>(A)</span> 1 <span class='latex-bold'>(A)</span>\ 1
<spanclass=′latex−bold′>(B)</span> 2 <span class='latex-bold'>(B)</span>\ \sqrt{2}
<spanclass=′latex−bold′>(C)</span> 1/2 <span class='latex-bold'>(C)</span>\ 1/\sqrt{2}
<spanclass=′latex−bold′>(D)</span> β does not attain a maximum value <span class='latex-bold'>(D)</span>\ \beta \text{ does not attain a maximum value}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #32: Suspended Pendulum

A thin, uniform rod has mass mm and length LL. Let the acceleration due to gravity be gg. Let the rotational inertia of the rod about its center be md2md^2.
The rod is suspended from a distance kdkd from the center, and undergoes small oscillations with an angular frequency βgd\beta \sqrt{\frac{g}{d}}.
Find an expression for β\beta in terms of kk.
<spanclass=′latex−bold′>(A)</span> 1+k2 <span class='latex-bold'>(A)</span>\ 1+k^2
<spanclass=′latex−bold′>(B)</span> 1+k2 <span class='latex-bold'>(B)</span>\ \sqrt{1+k^2}
<spanclass=′latex−bold′>(C)</span> k1+k <span class='latex-bold'>(C)</span>\ \sqrt{\frac{k}{1+k}}
<spanclass=′latex−bold′>(D)</span> k21+k <span class='latex-bold'>(D)</span>\ \sqrt{\frac{k^2}{1+k}}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #31: Rotational Inertia

A thin, uniform rod has mass mm and length LL. Let the acceleration due to gravity be gg. Let the rotational inertia of the rod about its center be md2md^2.
Find the ratio L/dL/d.
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 23<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(A)</span>\ 3\sqrt{2}\qquad<span class='latex-bold'>(B)</span>\ 3\qquad<span class='latex-bold'>(C)</span>\ 12\qquad<span class='latex-bold'>(D)</span>\ 2\sqrt{3}\qquad<span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #27: Ratio of Gravitational Fields

A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains NN astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in a gravitational field of strength gg. Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational field of strength g′g' . What is the ratio g′/gg'/g in terms of NN?[asy] import roundedpath; size(300); path a = roundedpath((0,-0.3)--(4,-0.3)--(4,-1)--(5,-1)--(5,0),0.1); draw(scale(+1,-1)*a); draw(scale(+1,+1)*a); draw(scale(-1,-1)*a); draw(scale(-1,+1)*a); filldraw(circle((0,0),1),white,black); filldraw(box((-2,-0.27),(2,0.27)),white,white); draw(arc((0,0),1.5,+35,+150),dashed,Arrow); draw(arc((0,0),1.5,-150,-35),dashed,Arrow);[/asy]
<spanclass=′latex−bold′>(A)</span> 2N/(N−1) <span class='latex-bold'>(A)</span>\ 2N/(N-1)
<spanclass=′latex−bold′>(B)</span> N/(N−1) <span class='latex-bold'>(B)</span>\ N/(N-1)
<spanclass=′latex−bold′>(C)</span> (N−1)/N <span class='latex-bold'>(C)</span>\ \sqrt{(N-1)/N}
<spanclass=′latex−bold′>(D)</span> N/(N−1) <span class='latex-bold'>(D)</span>\ \sqrt{N/(N-1)}
<spanclass=′latex−bold′>(E)</span> none of the above <span class='latex-bold'>(E)</span>\ \text{none of the above}
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2007 F = ma #19: Non-Hookian Spring

A non-Hookian spring has force F=−kx2F = -kx^2 where kk is the spring constant and xx is the displacement from its unstretched position. For the system shown of a mass mm connected to an unstretched spring initially at rest, how far does the spring extend before the system momentarily comes to rest? Assume that all surfaces are frictionless and that the pulley is frictionless as well.
[asy] size(250); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,-1)--(2,-1)--(2+sqrt(3),-2)); draw((2.5,-2)--(4.5,-2),dashed); draw(circle((2.2,-0.8),0.2)); draw((2.2,-0.8)--(1.8,-1.2)); draw((0,-0.6)--(0.6,-0.6)--(0.75,-0.4)--(0.9,-0.8)--(1.05,-0.4)--(1.2,-0.8)--(1.35,-0.4)--(1.5,-0.8)--(1.65,-0.4)--(1.8,-0.8)--(1.95,-0.6)--(2.2,-0.6)); draw((2+0.3*sqrt(3),-1.3)--(2+0.3*sqrt(3)+0.6/2,-1.3+sqrt(3)*0.6/2)--(2+0.3*sqrt(3)+0.6/2+0.2*sqrt(3),-1.3+sqrt(3)*0.6/2-0.2)--(2+0.3*sqrt(3)+0.2*sqrt(3),-1.3-0.2)); //super complex Asymptote code gg draw((2+0.3*sqrt(3)+0.3/2,-1.3+sqrt(3)*0.3/2)--(2.35,-0.6677)); draw(anglemark((2,-1),(2+sqrt(3),-2),(2.5,-2))); label("30∘30^\circ",(3.5,-2),NW); [/asy]
<spanclass=′latex−bold′>(A)</span> (3mg2k)1/2 <span class='latex-bold'>(A)</span>\ \left(\frac{3mg}{2k}\right)^{1/2}
<spanclass=′latex−bold′>(B)</span> (mgk)1/2 <span class='latex-bold'>(B)</span>\ \left(\frac{mg}{k}\right)^{1/2}
<spanclass=′latex−bold′>(C)</span> (2mgk)1/2 <span class='latex-bold'>(C)</span>\ \left(\frac{2mg}{k}\right)^{1/2}
<spanclass=′latex−bold′>(D)</span> (3mgk)1/3 <span class='latex-bold'>(D)</span>\ \left(\frac{\sqrt{3}mg}{k}\right)^{1/3}
<spanclass=′latex−bold′>(E)</span> (33mg2k)1/3 <span class='latex-bold'>(E)</span>\ \left(\frac{3\sqrt{3}mg}{2k}\right)^{1/3}
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2007 F = ma #25: Buoyant Oscillations

Find the period of small oscillations of a water pogo, which is a stick of mass m in the shape of a box (a rectangular parallelepiped.) The stick has a length LL, a width ww and a height hh and is bobbing up and down in water of density ρ\rho . Assume that the water pogo is oriented such that the length LL and width ww are horizontal at all times. Hint: The buoyant force on an object is given by Fbuoy=ρVgF_{buoy} = \rho Vg, where VV is the volume of the medium displaced by the object and ρ\rho is the density of the medium. Assume that at equilibrium, the pogo is floating.
<spanclass=′latex−bold′>(A)</span> 2πLg <span class='latex-bold'>(A)</span>\ 2\pi \sqrt{\frac{L}{g}}
<spanclass=′latex−bold′>(B)</span> πρw2L2gmh2 <span class='latex-bold'>(B)</span>\ \pi \sqrt{\frac{\rho w^2L^2 g}{mh^2}}
<spanclass=′latex−bold′>(C)</span> 2πmh2ρL2w2g <span class='latex-bold'>(C)</span>\ 2\pi \sqrt{\frac{mh^2}{\rho L^2w^2 g}}
<spanclass=′latex−bold′>(D)</span> 2πmρwLg<span class='latex-bold'>(D)</span>\ 2\pi \sqrt{\frac{m}{\rho wLg}}
<spanclass=′latex−bold′>(E)</span> πmρwLg<span class='latex-bold'>(E)</span>\ \pi \sqrt{\frac{m}{\rho wLg}}
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2007 F = ma #24: Angular Recoil

A ball of mass mm is launched into the air. Ignore air resistance, but assume that there is a wind that exerts a constant force F0F_0 in the -xx direction. In terms of F0F_0 and the acceleration due to gravity gg, at what angle above the positive xx-axis must the ball be launched in order to come back to the point from which it was launched?
<spanclass=′latex−bold′>(A)</span> tan⁡−1(F0/mg) <span class='latex-bold'>(A)</span>\ \tan^{-1}(F_0/mg)
<spanclass=′latex−bold′>(B)</span> tan⁡−1(mg/F0)<span class='latex-bold'>(B)</span>\ \tan^{-1}(mg/F_0)
<spanclass=′latex−bold′>(C)</span> sin⁡−1(F0/mg)<span class='latex-bold'>(C)</span>\ \sin^{-1}(F_0/mg)
<spanclass=′latex−bold′>(D)</span> the angle depends on the launch speed<span class='latex-bold'>(D)</span>\ \text{the angle depends on the launch speed}
<spanclass=′latex−bold′>(E)</span> no such angle is possible<span class='latex-bold'>(E)</span>\ \text{no such angle is possible}
23
1

2007 F = ma #23: Normal Force at the North Pole

If a planet of radius RR spins with an angular velocity ω\omega about an axis through the North Pole, what is the ratio of the normal force experienced by a person at the equator to that experienced by a person at the North Pole? Assume a constant gravitational field gg and that both people are stationary relative to the planet and are at sea level.
<spanclass=′latex−bold′>(A)</span> g/Rω2 <span class='latex-bold'>(A)</span>\ g/R\omega^2
<spanclass=′latex−bold′>(B)</span> Rω2/g<span class='latex-bold'>(B)</span>\ R\omega^2/g
<spanclass=′latex−bold′>(C)</span> 1−Rω2/g<span class='latex-bold'>(C)</span>\ 1- R\omega^2/g
<spanclass=′latex−bold′>(D)</span> 1+g/Rω2<span class='latex-bold'>(D)</span>\ 1+g/R\omega^2
<spanclass=′latex−bold′>(E)</span> 1+Rω2/g<span class='latex-bold'>(E)</span>\ 1+R\omega^2/g
22
1

2007 F = ma #22: Rocket Momentum

Two rockets are in space in a negligible gravitational field. All observations are made by an observer in a reference frame in which both rockets are initially at rest. The masses of the rockets are mm and 9m9m. A constant force FF acts on the rocket of mass m for a distance dd. As a result, the rocket acquires a momentum pp. If the same constant force FF acts on the rocket of mass 9m9m for the same distance dd, how much momentum does the rocket of mass 9m9m acquire?
<spanclass=′latex−bold′>(A)</span> p/9<spanclass=′latex−bold′>(B)</span> p/3<spanclass=′latex−bold′>(C)</span> p<spanclass=′latex−bold′>(D)</span> 3p<spanclass=′latex−bold′>(E)</span> 9p <span class='latex-bold'>(A)</span>\ p/9 \qquad<span class='latex-bold'>(B)</span>\ p/3 \qquad<span class='latex-bold'>(C)</span>\ p \qquad<span class='latex-bold'>(D)</span>\ 3p \qquad<span class='latex-bold'>(E)</span>\ 9p
21
1

2007 F = ma #21: Rotational Inertia

If the rotational inertia of a sphere about an axis through the center of the sphere is II, what is the rotational inertia of another sphere that has the same density, but has twice the radius?
<spanclass=′latex−bold′>(A)</span> 2I<spanclass=′latex−bold′>(B)</span> 4I<spanclass=′latex−bold′>(C)</span> 8I<spanclass=′latex−bold′>(D)</span> 16I<spanclass=′latex−bold′>(E)</span> 32I <span class='latex-bold'>(A)</span>\ 2I \qquad<span class='latex-bold'>(B)</span>\ 4I \qquad<span class='latex-bold'>(C)</span>\ 8I\qquad<span class='latex-bold'>(D)</span>\ 16I\qquad<span class='latex-bold'>(E)</span>\ 32I
20
1

2007 F = ma #20: Wall Collisions

A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy EE. With every collision with the walls, the mass loses 1/21/2 its kinetic energy to thermal energy. How many collisions with the walls are necessary before the speed of the mass is reduced by a factor of 88?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 6<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 16 <span class='latex-bold'>(A)</span>\ 3\qquad<span class='latex-bold'>(B)</span>\ 4\qquad<span class='latex-bold'>(C)</span>\ 6\qquad<span class='latex-bold'>(D)</span>\ 8\qquad<span class='latex-bold'>(E)</span>\ 16
18
1

2007 F = ma #18: A Small Chunk of Ice

A small chunk of ice falls from rest down a frictionless parabolic ice sheet shown in the figure. At the point labeled A\mathbf{A} in the diagram, the ice sheet becomes a steady, rough incline of angle 30∘30^\circ with respect to the horizontal and friction coefficient μk\mu_k. This incline is of length 32h\frac{3}{2}h and ends at a cliff. The chunk of ice comes to a rest precisely at the end of the incline. What is the coefficient of friction μk\mu_k?
[asy] size(200); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(sqrt(3),0)--(0,1)); draw(anglemark((0,1),(sqrt(3),0),(0,0))); label("30∘30^\circ",(1.5,0.03),NW); label("A", (0,1),NE); dot((0,1)); label("rough incline",(0.4,0.4)); draw((0.4,0.5)--(0.5,0.6),EndArrow); dot((-0.2,4/3)); label("parabolic ice sheet",(0.6,4/3)); draw((0.05,1.3)--(-0.05,1.2),EndArrow); label("ice chunk",(-0.5,1.6)); draw((-0.3,1.5)--(-0.25,1.4),EndArrow); draw((-0.2,4/3)--(-0.19, 1.30083)--(-0.18,1.27)--(-0.17,1.240833)--(-0.16,1.21333)--(-0.15,1.1875)--(-0.14,1.16333)--(-0.13,1.140833)--(-0.12,1.12)--(-0.11,1.100833)--(-0.10,1.08333)--(-0.09,1.0675)--(-0.08,1.05333)--(-0.07,1.040833)--(-0.06,1.03)--(-0.05,1.020833)--(-0.04,1.01333)--(-0.03,1.0075)--(-0.02,1.00333)--(-0.01,1.000833)--(0,1)); draw((-0.6,0)--(-0.6,4/3),dashed,EndArrow,BeginArrow); label("hh",(-0.6,2/3),W); draw((0.2,1.2)--(sqrt(3)+0.2,0.2),dashed,EndArrow,BeginArrow); label("32h\frac{3}{2}h",(sqrt(3)/2+0.2,0.7),NE); [/asy]
<spanclass=′latex−bold′>(A)</span> 0.866<spanclass=′latex−bold′>(B)</span> 0.770<spanclass=′latex−bold′>(C)</span> 0.667<spanclass=′latex−bold′>(D)</span> 0.385<spanclass=′latex−bold′>(E)</span> 0.333 <span class='latex-bold'>(A)</span>\ 0.866\qquad<span class='latex-bold'>(B)</span>\ 0.770\qquad<span class='latex-bold'>(C)</span>\ 0.667\qquad<span class='latex-bold'>(D)</span>\ 0.385\qquad<span class='latex-bold'>(E)</span>\ 0.333
17
1

2007 F = ma #17: Tangential Component of Acceleration

A small point-like object is thrown horizontally off of a 50.050.0-m\text{m} high building with an initial speed of 10.0 m/s10.0 \text{ m/s}. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object? Ignore air resistance.
<spanclass=′latex−bold′>(A)</span> 2.00 s <span class='latex-bold'>(A)</span>\ 2.00\text{ s}
<spanclass=′latex−bold′>(B)</span> 1.50 s<span class='latex-bold'>(B)</span>\ 1.50\text{ s}
<spanclass=′latex−bold′>(C)</span> 1.00 s<span class='latex-bold'>(C)</span>\ 1.00\text{ s}
<spanclass=′latex−bold′>(D)</span> 0.50 s<span class='latex-bold'>(D)</span>\ 0.50\text{ s}
<spanclass=′latex−bold′>(E)</span> The building is not high enough for this to occur.<span class='latex-bold'>(E)</span>\ \text{The building is not high enough for this to occur.}
16
1

2007 F = ma #16: Baseball/Basketball Collision

A baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of 4.0 m/s4.0 \text{ m/s}, and collides with the baseball as it is moving downward at 4.0 m/s4.0 \text{ m/s}. After the collision, the baseball moves upward as shown in the figure and the basketball is instantaneously at rest right after the collision. The mass of the baseball is 0.2 kg0.2 \text{ kg} and the mass of the basketball is 0.5 kg0.5 \text{ kg}. Ignore air resistance and ignore any changes in velocities due to gravity during the very short collision times. The speed of the baseball right after colliding with the upward moving basketball is
[asy] size(200); path P=CR((0,0),1); picture a; pen p=gray(0.5)+linewidth(1.5); fill(a,P,gray(0.8)); draw(a,arc((0,0),0.6,30,240),p); draw(a,arc(1.2*dir(30),0.6,210,360),p); draw(a,arc(1.2*dir(240),0.6,-170,60),p); clip(a,P); real t=17; draw((0,t+1)--(0,t+6),linewidth(1),EndArrow(size=7)); add(shift((0,t))*a); fill(a,P,gray(0.8)); draw(a,(-1,-1)--(1,1),p); draw(a,arc(dir(-45),0.8,0,330),p); draw(a,arc(dir(135),0.8,-160,180),p); draw(a,0.2*dir(-45)--dir(-45)^^0.2*dir(135)--dir(135),p); clip(a,P); add(scale(4)*a); path Q=xscale(12)*yscale(0.5)*unitsquare; draw(shift((-6,-6))*Q,p); draw(shift((-6,-6.5))*Q,p);[/asy]
<spanclass=′latex−bold′>(A)</span> 4.0 m/s<spanclass=′latex−bold′>(B)</span> 6.0 m/s<spanclass=′latex−bold′>(C)</span> 8.0 m/s<spanclass=′latex−bold′>(D)</span> 12.0 m/s<spanclass=′latex−bold′>(E)</span> 16.0 m/s <span class='latex-bold'>(A)</span>\ 4.0\text{ m/s}\qquad<span class='latex-bold'>(B)</span>\ 6.0\text{ m/s}\qquad<span class='latex-bold'>(C)</span>\ 8.0\text{ m/s}\qquad<span class='latex-bold'>(D)</span>\ 12.0\text{ m/s}\qquad<span class='latex-bold'>(E)</span>\ 16.0\text{ m/s}
15
1

2007 F = ma #15: Pulley Tension

A uniform disk (I=12MR2 I = \dfrac {1}{2} MR^2 ) of mass 8.0 kg can rotate without friction on a fixed axis. A string is wrapped around its circumference and is attached to a 6.0 kg mass. The string does not slip. What is the tension in the cord while the mass is falling? [asy] size(250); pen p=linewidth(3), dg=gray(0.25), llg=gray(0.90), lg=gray(0.75),g=grey; void f(path P, pen p, pen q) { filldraw(P,p,q); } path P=CR((0,0),1); D((1,0)--(1,-2.5),p+lg); f(P,g,p); P=scale(0.4)*P; f(P,lg,p); path Q=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; f(Q,llg,dg+p); P=scale(0.45)*P; f(P,llg,dg+p); P=shift((0.15,0.15))*((-1,-1)--(-1,-2)--(-1.1,-2)--(-1.1,-1.1)--(-2,-1.1)--(-2,-1)--cycle); f(P,llg,lg+p); P=shift((1.55,1.55))*scale(3)*P; f(P,llg,g+p); unfill((-1.23,-1.23)--(-1.23,-5)--(-5,-1.23)--cycle); clip((-3.8,-3.8)--(-3.8,3.8)--(3.8,3.8)--(3.8,-3.8)--cycle); P=(0.2,-2.5)--(1.8,-2.5)--(1.8,-4.1)--(0.2,-4.1)--cycle; f(P,llg,lg+p); MP("m",(1,-3.3),(0,0),fontsize(16)); MP("M",(0,-1),fontsize(16));[/asy] (A) 20.0 N(B) 24.0 N(C) 34.3 N(D) 60.0 N(E) 80.0 N \textbf {(A) } \text {20.0 N} \qquad \textbf {(B) } \text {24.0 N} \qquad \textbf {(C) } \text {34.3 N} \qquad \textbf {(D) } \text {60.0 N} \qquad \textbf {(E) } \text {80.0 N}
14
1

2007 F = ma #14: Friction of a Rear-Drive Car

When the speed of a rear-drive car is increasing on a horizontal road, the direction of the frictional force on the tires is:
(A) backward on the front tires and forward on the rear tires. \textbf {(A) } \text {backward on the front tires and forward on the rear tires.}
(B) forward on the front tires and backward on the rear tires. \textbf {(B) } \text {forward on the front tires and backward on the rear tires.}
(C) forward on all tires. \textbf {(C) } \text {forward on all tires.}
(D) backward on all tires. \textbf {(D) } \text {backward on all tires.}
(E) zero. \textbf {(E) } \text {zero.}
13
1

2007 F = ma #13: Elastic Collision Graph

A particle moves along the xx-axis. It collides elastically head-on with an identical particle initially at rest. Which of the following graphs could illustrate the momentum of each particle as a function of time?
[asy] size(400); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,5)); draw((0,1.5)--(5,1.5)); label("pp",(0,5),N); label("tt",(5,1.5),E); label("(A)\mathbf{(A)}",(2.5,-0.5)); draw((0,1.5)--(2.5,1.5)--(2.5,0.75)--(4,0.75),black+linewidth(2)); draw((0,3.5)--(2.5,3.5)--(2.5,4.25)--(4,4.25),black+linewidth(2));
draw((8,0)--(8,5)); draw((8,1.5)--(13,1.5)); label("pp",(8,5),N); label("tt",(13,1.5),E); label("(B)\mathbf{(B)}",(10.5,-0.5)); draw((8,1.5)--(10.5,1.5)--(10.5,2.5)--(12,2.5),black+linewidth(2)); draw((8,3.5)--(10.5,3.5)--(10.5,4.5)--(12,4.5),black+linewidth(2));
draw((16,0)--(16,5)); draw((16,1.5)--(21,1.5)); label("pp",(16,5),N); label("tt",(21,1.5),E); label("(C)\mathbf{(C)}",(18.5,-0.5)); draw((16,1.5)--(18.5,1.5)--(18.5,2.25)--(20,2.25),black+linewidth(2)); draw((16,3.5)--(18.5,3.5)--(18.5,2.75)--(20,2.75),black+linewidth(2));
draw((24,0)--(24,5)); draw((24,1.5)--(29,1.5)); label("pp",(24,5),N); label("tt",(29,1.5),E); label("(D)\mathbf{(D)}",(26.5,-0.5)); draw((24,1.5)--(26.5,1.5)--(26.75,3.25)--(28,3.25),black+linewidth(2)); draw((24,3.25)--(26.5,3.25)--(26.75,1.5)--(28,1.5),black+linewidth(2)); draw((32,0)--(32,5)); draw((32,1.5)--(37,1.5)); label("pp",(32,5),N); label("tt",(37,1.5),E); label("(E)\mathbf{(E)}",(34.5,-0.5)); draw((32,1.5)--(34.5,1.5)--(34.5,0.5)--(36,0.5),black+linewidth(2)); draw((32,3.5)--(34.5,3.5)--(34.5,2.75)--(36,2.75),black+linewidth(2)); [/asy]
12
1

2007 F = ma #12: Rock hanging on a seesaw

A 22-kg rock is suspended by a massless string from one end of a uniform 11-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the 0.200.20-meter mark?
[asy] size(250); draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle); draw((1.5,0)--(1.5,0.2)); draw((3,0)--(3,0.2)); draw((4.5,0)--(4.5,0.2)); draw((6,0)--(6,0.2)); filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8)); draw((0,0)--(0,-0.4)); filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4));
[/asy]
(A) 0.20 kg(B) 1.00 kg(C) 1.33 kg(D) 2.00 kg(E) 3.00 kg \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg}
11
1

2007 F = ma #11: Kinetic Energy Comparison

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time tt, from least to greatest.
[asy] size(225); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.7)); draw((0,-1)--(2,-1),EndArrow); label("F⃗\vec{F}",(1, -1),S); label("Disk",(-1,0),W); filldraw(circle((5,0),1),gray(.7)); filldraw(circle((5,0),0.75),white); draw((5,-1)--(7,-1),EndArrow); label("F⃗\vec{F}",(6, -1),S); label("Hoop",(6,0),E); filldraw(circle((10,0),1),gray(.5)); draw((10,-1)--(12,-1),EndArrow); label("F⃗\vec{F}",(11, -1),S); label("Sphere",(11,0),E); [/asy]
<spanclass=′latex−bold′>(A)</span> disk, hoop, sphere <span class='latex-bold'>(A)</span> \ \text{disk, hoop, sphere}
<spanclass=′latex−bold′>(B)</span> sphere, disk, hoop<span class='latex-bold'>(B)</span>\ \text{sphere, disk, hoop}
<spanclass=′latex−bold′>(C)</span> hoop, sphere, disk<span class='latex-bold'>(C)</span>\ \text{hoop, sphere, disk}
<spanclass=′latex−bold′>(D)</span> disk, sphere, hoop<span class='latex-bold'>(D)</span>\ \text{disk, sphere, hoop}
<spanclass=′latex−bold′>(E)</span> hoop, disk, sphere<span class='latex-bold'>(E)</span>\ \text{hoop, disk, sphere}
10
1

2007 F = ma #10: Identical Angular Accelerations

Two wheels with fixed hubs, each having a mass of 1 kg1 \text{ kg}, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is I=mR2I = mR^2. In order to impart identical angular accelerations about their respective hubs, how large must F2F_2 be?
[asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw(circle((0,0),0.5)); draw((1, 0.5)--(0,0.5)--(0,-0.5),BeginArrow); draw((-0.5,0)--(0.5,0)); draw((-0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(0.5*sqrt(2)/2,-0.5*sqrt(2)/2)); draw((0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(-0.5*sqrt(2)/2,-0.5*sqrt(2)/2)); label("RR = 0.5 m", (0, -0.5),S); label("F1F_1 = 1 N",(1,0.5),N); draw(circle((3,0.5),1)); draw((4.5,1.5)--(3,1.5)--(3,-0.5),BeginArrow); draw((2,0.5)--(4,0.5)); draw((3-sqrt(2)/2, 0.5+sqrt(2)/2)--(3+sqrt(2)/2, 0.5-sqrt(2)/2)); draw((3+sqrt(2)/2, 0.5+sqrt(2)/2)--(3-sqrt(2)/2,0.5-sqrt(2)/2)); label("F2F_2", (4.5, 1.5), N); label("RR = 1 m",(3, -0.5),S); [/asy]
<spanclass=′latex−bold′>(A)</span> 0.25 N<spanclass=′latex−bold′>(B)</span> 0.5 N<spanclass=′latex−bold′>(C)</span> 1 N<spanclass=′latex−bold′>(D)</span> 2 N<spanclass=′latex−bold′>(E)</span> 4 N <span class='latex-bold'>(A)</span>\ 0.25\text{ N}\qquad<span class='latex-bold'>(B)</span>\ 0.5\text{ N}\qquad<span class='latex-bold'>(C)</span>\ 1\text{ N}\qquad<span class='latex-bold'>(D)</span>\ 2\text{ N}\qquad<span class='latex-bold'>(E)</span>\ 4\text{ N}
9
1

2007 F = ma #9: Movement of Center of Mass

A large wedge rests on a horizontal frictionless surface, as shown. A block starts from rest and slides down the inclined surface of the wedge, which is rough. During the motion of the block, the center of mass of the block and wedge
[asy]
draw((0,0)--(10,0),linewidth(1)); filldraw((2.5,0)--(6.5,2.5)--(6.5,0)--cycle, gray(.9),linewidth(1)); filldraw((5, 12.5/8)--(6,17.5/8)--(6-5/8, 17.5/8+1)--(5-5/8,12.5/8+1)--cycle, gray(.2)); [/asy]
<spanclass=′latex−bold′>(A)</span> does not move<span class='latex-bold'>(A)</span>\ \text{does not move}
<spanclass=′latex−bold′>(B)</span> moves horizontally with constant speed<span class='latex-bold'>(B)</span>\ \text{moves horizontally with constant speed}
<spanclass=′latex−bold′>(C)</span> moves horizontally with increasing speed<span class='latex-bold'>(C)</span>\ \text{moves horizontally with increasing speed}
<spanclass=′latex−bold′>(D)</span> moves vertically with increasing speed<span class='latex-bold'>(D)</span>\ \text{moves vertically with increasing speed}
<spanclass=′latex−bold′>(E)</span> moves both horizontally and vertically<span class='latex-bold'>(E)</span>\ \text{moves both horizontally and vertically}
8
1

2007 F = ma #8: Gravitational Potential Energy

When two stars are very far apart their gravitational potential energy is zero; when they are separated by a distance dd the gravitational potential energy of the system is UU. If the stars are separated by a distance 2d2d the gravitational potential energy of the system is
<spanclass=′latex−bold′>(A)</span> U/4<spanclass=′latex−bold′>(B)</span> U/2<spanclass=′latex−bold′>(C)</span> U<spanclass=′latex−bold′>(D)</span> 2U<spanclass=′latex−bold′>(E)</span> 4U <span class='latex-bold'>(A)</span>\ U/4\qquad<span class='latex-bold'>(B)</span>\ U/2 \qquad<span class='latex-bold'>(C)</span>\ U \qquad<span class='latex-bold'>(D)</span>\ 2U\qquad<span class='latex-bold'>(E)</span>\ 4U
7
1

2007 F = ma #7: Chemical Potential Energy

The chemical potential energy stored in a battery is converted into kinetic energy in a toy car that increases its speed first from 0 mph0 \text{ mph} to 2 mph2 \text{ mph} and then from 2 mph2 \text{ mph} up to 4 mph4 \text{ mph}. Ignore the energy transferred to thermal energy due to friction and air resistance. Compared to the energy required to go from 00 to 2 mph2 \text{ mph}, the energy required to go from 22 to 4 mph4 \text{ mph} is
<spanclass=′latex−bold′>(A)</span> half the amount. <span class='latex-bold'>(A)</span>\ \text{half the amount.}
<spanclass=′latex−bold′>(B)</span> the same amount. <span class='latex-bold'>(B)</span>\ \text{the same amount.}
<spanclass=′latex−bold′>(C)</span> twice the amount. <span class='latex-bold'>(C)</span>\ \text{twice the amount.}
<spanclass=′latex−bold′>(D)</span> three times the amount. <span class='latex-bold'>(D)</span>\ \text{three times the amount.}
<spanclass=′latex−bold′>(E)</span> four times the amount. <span class='latex-bold'>(E)</span>\ \text{four times the amount.}
6
1

2007 F = ma #6: Displacement of Drag Racer

At time t=0t = 0 a drag racer starts from rest at the origin and moves along a straight line with velocity given by v=5t2v = 5t^2, where vv is in m/s\text{m/s} and tt in s\text{s}. The expression for the displacement of the car from t=0t = 0 to time tt is
<spanclass=′latex−bold′>(A)</span> 5t3<spanclass=′latex−bold′>(B)</span> 5t3/3<spanclass=′latex−bold′>(C)</span> 10t<spanclass=′latex−bold′>(D)</span> 15t2<spanclass=′latex−bold′>(E)</span> 5t/2 <span class='latex-bold'>(A)</span>\ 5t^3 \qquad<span class='latex-bold'>(B)</span>\ 5t^3/3\qquad<span class='latex-bold'>(C)</span>\ 10t \qquad<span class='latex-bold'>(D)</span>\ 15t^2 \qquad<span class='latex-bold'>(E)</span>\ 5t/2
5
1

2007 F = ma #5: Crate of Toys

A crate of toys remains at rest on a sleigh as the sleigh is pulled up a hill with an increasing speed. The crate is not fastened down to the sleigh. What force is responsible for the crate’s increase in speed up the hill?
<spanclass=′latex−bold′>(A)</span> the force of static friction of the sleigh on the crate<span class='latex-bold'>(A)</span> \ \text{the force of static friction of the sleigh on the crate}
<spanclass=′latex−bold′>(B)</span> the contact force (normal force) of the ground on the sleigh <span class='latex-bold'>(B)</span> \ \text{the contact force (normal force) of the ground on the sleigh}
<spanclass=′latex−bold′>(C)</span> the contact force (normal force) of the sleigh on the crate <span class='latex-bold'>(C)</span> \ \text{the contact force (normal force) of the sleigh on the crate}
<spanclass=′latex−bold′>(D)</span> the gravitational force acting on the sleigh <span class='latex-bold'>(D)</span> \ \text{the gravitational force acting on the sleigh}
<spanclass=′latex−bold′>(E)</span> no force is needed <span class='latex-bold'>(E)</span> \ \text{no force is needed}
4
1

2007 F = ma #4: Fall Distance

An object is released from rest and falls a distance hh during the first second of time. How far will it fall during the next second of time?
<spanclass=′latex−bold′>(A)</span> h<spanclass=′latex−bold′>(B)</span> 2h<spanclass=′latex−bold′>(C)</span> 3h<spanclass=′latex−bold′>(D)</span> 4h<spanclass=′latex−bold′>(E)</span> h2 <span class='latex-bold'>(A)</span>\ h\qquad<span class='latex-bold'>(B)</span>\ 2h \qquad<span class='latex-bold'>(C)</span>\ 3h \qquad<span class='latex-bold'>(D)</span>\ 4h\qquad<span class='latex-bold'>(E)</span>\ h^2
3
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2007 F = ma #3: Average Velocity

The coordinate of an object is given as a function of time by x=8t−3t2x = 8t - 3t^2, where xx is in meters and tt is in seconds. Its average velocity over the interval from t=1 t = 1 to t=2 st = 2 \text{ s} is
<spanclass=′latex−bold′>(A)</span> −2 m/s<spanclass=′latex−bold′>(B)</span> −1 m/s<spanclass=′latex−bold′>(C)</span> −0.5 m/s<spanclass=′latex−bold′>(D)</span> 0.5 m/s<spanclass=′latex−bold′>(E)</span> 1 m/s <span class='latex-bold'>(A)</span>\ -2\text{ m/s}\qquad<span class='latex-bold'>(B)</span>\ -1\text{ m/s}\qquad<span class='latex-bold'>(C)</span>\ -0.5\text{ m/s}\qquad<span class='latex-bold'>(D)</span>\ 0.5\text{ m/s}\qquad<span class='latex-bold'>(E)</span>\ 1\text{ m/s}
2
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2007 F = ma #2: Car Velocity

The graph shows velocity as a function of time for a car. What was the acceleration at time = 9090 seconds?
[asy] size(275); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2)); draw((0,3)--(6,3)); draw((0,4)--(6,4)); draw((0,0)--(0,4)); draw((1,0)--(1,4)); draw((2,0)--(2,4)); draw((3,0)--(3,4)); draw((4,0)--(4,4)); draw((5,0)--(5,4)); draw((6,0)--(6,4)); label("00",(0,0),S); label("3030",(1,0),S); label("6060",(2,0),S); label("9090",(3,0),S); label("120120",(4,0),S); label("150150",(5,0),S); label("180180",(6,0),S); label("00",(0,0),W); label("1010",(0,1),W); label("2020",(0,2),W); label("3030",(0,3),W); label("4040",(0,4),W); draw((0,0.6)--(0.1,0.55)--(0.8,0.55)--(1.2,0.65)--(1.9,1)--(2.2,1.2)--(3,2)--(4,3)--(4.45,3.4)--(4.5,3.5)--(4.75,3.7)--(5,3.7)--(5.5,3.45)--(6,3)); label("Time (s)", (7.5,0),S); label("Velocity (m/s)",(-1,3),W); [/asy]
<spanclass=′latex−bold′>(A)</span> 0.2 m/s2<spanclass=′latex−bold′>(B)</span> 0.33 m/s2<spanclass=′latex−bold′>(C)</span> 1.0 m/s2<spanclass=′latex−bold′>(D)</span> 9.8 m/s2<spanclass=′latex−bold′>(E)</span> 30 m/s2 <span class='latex-bold'>(A)</span>\ 0.2\text{ m/s}^2\qquad<span class='latex-bold'>(B)</span>\ 0.33\text{ m/s}^2\qquad<span class='latex-bold'>(C)</span>\ 1.0\text{ m/s}^2\qquad<span class='latex-bold'>(D)</span>\ 9.8\text{ m/s}^2\qquad<span class='latex-bold'>(E)</span>\ 30\text{ m/s}^2
1
1

2007 F = ma #1: Position Vector

An object moves in two dimensions according to r⃗(t)=(4.0t2−9.0)i⃗+(2.0t−5.0)j⃗\vec{r}(t) = (4.0t^2 - 9.0)\vec{i} + (2.0t - 5.0)\vec{j} where rr is in meters and tt in seconds. When does the object cross the xx-axis?
<spanclass=′latex−bold′>(A)</span> 0.0 s<spanclass=′latex−bold′>(B)</span> 0.4 s<spanclass=′latex−bold′>(C)</span> 0.6 s<spanclass=′latex−bold′>(D)</span> 1.5 s<spanclass=′latex−bold′>(E)</span> 2.5 s <span class='latex-bold'>(A)</span>\ 0.0 \text{ s}\qquad<span class='latex-bold'>(B)</span>\ 0.4 \text{ s}\qquad<span class='latex-bold'>(C)</span>\ 0.6 \text{ s}\qquad<span class='latex-bold'>(D)</span>\ 1.5 \text{ s}\qquad<span class='latex-bold'>(E)</span>\ 2.5 \text{ s}