A rocket moves in deep space with negligible external force. Its exhaust speed relative to the rocket is u. Its mass decreases from $M_0$ to M. The increase in the rocket's speed is:
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Question

A rocket moves in deep space with negligible external force. Its exhaust speed relative to the rocket is u. Its mass decreases from M 0 M_0 M 0 ​ to M. The increase in the rocket's speed is:

G1 Physics · Accepted Answer

For a rocket with exhaust speed u relative to the rocket and no external force, the differential rocket equation is M dv = -u dM. As the rocket loses mass, dM is negative, so dv is positive. Integrate from initial mass M0 and speed v0 to final mass M and speed v: ∫(v0 to v) dv = -u∫(M0 to M) dM/M. Thus v-v0 = -u ln(M/M0) = u ln(M0/M). Therefore the increase in speed is Δv = u ln(M0/M). Correct option: A.

A rocket moves in deep space with negligible external force. Its exhaust speed relative to the rocket is u. Its mass decreases from M 0 M_0 M 0 ​ to M. The increase in the rocket's speed is:

A rocket moves in deep space with negligible external force. Its exhaust speed relative to the rocket is u. Its mass decreases from M 0 M_0 M 0 to M. The increase in the rocket's speed is: