F=ma 2024 Problem 18

Physics · mechanics

Problem

Question: As shown in the figure, a satellite is initially in a circular orbit of radius

R

around a planet of mass

M

. It fires its rockets to instantaneously increase its speed by

\Delta v

, keeping the direction of its velocity the same, so that it enters an elliptical orbit whose maximum distance from the planet is

2R

. What is the value of

\Delta v

? (Hint: when the satellite is in an elliptical orbit with semimajor axis

a

, its total energy per unit mass is

- {GM}/{2a}

.) (A)

0.08 \sqrt{\frac{GM}{R}}

. (B)

0.15 \sqrt{\frac{GM}{R}}

. (C)

0.22 \sqrt{\frac{GM}{R}}

. (D)

0.29 \sqrt{\frac{GM}{R}}

. (E)

0.41 \sqrt{\frac{GM}{R}}

. *Note: This problem references figure(s): `image_question/F=MA_2024_18_1.png`*