MathDB

Spaceman Fred's spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is

R

; the maximum distance between the spaceship and the planet is

2R

. At the point of maximum distance, Spaceman Fred is traveling at speed

v_\text{0}

. He then fires his thrusters so that he enters a circular orbit of radius

2R

. What is his new speed? [asy] size(300); // Shape draw(circle((0,0),25),dashed+gray); draw(circle((0,0),3.5),linewidth(2)); draw(ellipse((5,0),20,15)); // Dashed Lines draw((25,13)--(25,-35),dotted); draw((0,-35)--(0,-3.3),dotted); draw((0,3.3)--(0,13),dotted); draw((-15,13)--(-15,-35),dotted); // Labels draw((-14,-35)--(-1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"

R

",(-7.5,-35),N); draw((24,-35)--(1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"

2R

",(10,-35),N); // Blobs on Earth path A=(-1.433, 2.667)-- (-1.433, 2.573)-- (-1.360, 2.478)-- (-1.408, 2.360)-- (-1.493, 2.207)-- (-1.554, 2.160)-- (-1.614, 2.113)-- (-1.675, 2.065)-- (-1.735, 1.959)-- (-1.772, 1.877)-- (-1.723, 1.759)-- (-1.748, 1.676)-- (-1.748, 1.523)-- (-1.772, 1.369)-- (-1.760, 1.240)-- (-1.857, 1.145)-- (-1.941, 1.098)-- (-2.050, 1.122)-- (-2.111, 1.086)-- (-2.244, 1.039)-- (-2.390, 1.004)-- (-2.511, 0.909)-- (-2.486, 0.697)-- (-2.499, 0.555)-- (-2.535, 0.414)-- (-2.668, 0.308)-- (-2.765, 0.237)-- (-2.910, 0.131)-- (-3.068, 0.036)-- (-3.250, 0.024)-- (-3.310, 0.154)-- (-3.274, 0.272)-- (-3.286, 0.402)-- (-3.298, 0.532)-- (-3.250, 0.650)-- (-3.165, 0.768)-- (-3.128, 0.933)-- (-3.068, 1.074)-- (-3.032, 1.204)-- (-2.971, 1.310)-- (-2.886, 1.452)-- (-2.801, 1.558)-- (-2.729, 1.652)-- (-2.656, 1.770)-- (-2.583, 1.912)-- (-2.486, 1.995)-- (-2.365, 2.089)-- (-2.244, 2.207)-- (-2.123, 2.313)-- (-2.014, 2.419)-- (-1.905, 2.478)-- (-1.832, 2.573)-- (-1.687, 2.643)-- (-1.578, 2.714)--cycle; filldraw(A,gray); path B=(-0.397, 2.527)-- (-0.468, 2.321)-- (-0.538, 2.154)-- (-0.639, 2.065)-- (-0.760, 2.085)-- (-0.922, 2.085)-- (-0.993, 2.016)-- (-0.770, 1.918)-- (-0.649, 1.829)-- (-0.498, 1.780)-- (-0.367, 1.770)-- (-0.205, 1.751)-- (-0.084, 1.761)-- (-0.104, 1.613)-- (-0.114, 1.495)-- (-0.094, 1.358)-- (0.007, 1.220)-- (0.067, 1.131)-- (0.108, 1.013)-- (0.188, 0.905)-- (0.239, 0.787)-- (0.330, 0.650)-- (0.461, 0.620)-- (0.622, 0.620)-- (0.794, 0.591)-- (0.905, 0.610)-- (0.956, 0.689)-- (1.026, 0.591)-- (1.097, 0.483)-- (1.198, 0.374)-- (1.258, 0.276)-- (1.339, 0.188)-- (1.319, -0.009)-- (1.309, -0.166)-- (1.198, -0.343)-- (1.077, -0.432)-- (0.935, -0.520)-- (0.814, -0.589)-- (0.633, -0.677)-- (0.481, -0.727)-- (0.350, -0.776)-- (0.229, -0.894)-- (0.229, -1.041)-- (0.229, -1.228)-- (0.340, -1.346)-- (0.522, -1.415)-- (0.643, -1.513)-- (0.693, -1.651)-- (0.784, -1.798)-- (0.723, -1.936)-- (0.612, -2.044)-- (0.471, -2.123)-- (0.350, -2.201)-- (0.249, -2.270)-- (0.108, -2.339)-- (-0.013, -2.418)-- (-0.124, -2.535)-- (-0.135, -2.673)-- (-0.175, -2.811)-- (-0.084, -2.840)-- (0.067, -2.840)-- (0.209, -2.830)-- (0.350, -2.742)-- (0.522, -2.653)-- (0.582, -2.604)-- (0.713, -2.545)-- (0.845, -2.457)-- (0.935, -2.408)-- (1.057, -2.388)-- (1.228, -2.280)-- (1.329, -2.191)-- (1.460, -2.132)-- (1.581, -2.093)-- (1.692, -2.044)-- (1.793, -2.005)-- (1.844, -1.906)-- (1.844, -1.828)-- (1.904, -1.749)-- (2.005, -1.621)-- (1.955, -1.454)-- (1.894, -1.287)-- (1.773, -1.189)-- (1.632, -0.992)-- (1.592, -0.874)-- (1.491, -0.736)-- (1.410, -0.569)-- (1.460, -0.412)-- (1.561, -0.274)-- (1.592, -0.078)-- (1.622, 0.168)-- (1.551, 0.306)-- (1.440, 0.404)-- (1.420, 0.561)-- (1.551, 0.620)-- (1.703, 0.630)-- (1.824, 0.532)-- (1.955, 0.365)-- (2.046, 0.453)-- (2.116, 0.551)-- (2.167, 0.689)-- (2.096, 0.807)-- (1.965, 0.905)-- (1.834, 0.935)-- (1.743, 0.994)-- (1.622, 1.131)-- (1.531, 1.249)-- (1.430, 1.348)-- (1.359, 1.515)-- (1.420, 1.702)-- (1.511, 1.839)-- (1.571, 2.016)-- (1.672, 2.134)-- (1.592, 2.232)-- (1.440, 2.291)-- (1.289, 2.350)-- (1.178, 2.252)-- (1.127, 2.134)-- (1.067, 1.997)-- (0.986, 1.898)-- (0.845, 1.839)-- (0.693, 1.839)-- (0.522, 1.859)-- (0.471, 1.977)-- (0.380, 2.124)-- (0.289, 2.203)-- (0.188, 2.291)-- (0.047, 2.311)-- (-0.074, 2.370)-- (-0.195, 2.508)--cycle; filldraw(B,gray); [/asy] (A)

\sqrt{3/2}v_\text{0}

(B)

\sqrt{5}v_\text{0}

(C)

\sqrt{3/5}v_\text{0}

(D)

\sqrt{2}v_\text{0}

(E)

2v_\text{0}

25

Problems(1)