Physics Problems & Solutions: Classical Mechanics

Questions 61-62

The Equation of motion of a rocket in free space can be written where m is the rocket’s mass, v is its velocity, t is time, and u is a constant.

$m \frac{dv}{dt}+u \frac{dm}{dt}=0$

The constant u represents the speed of the

A. Rocket at t = 0
B. Rocket after its fuel is spent
C. Rocket in its instantaneous rest frame
D. Rocket’s exhaust in a stationary frame
E. Rocket’s exhaust relative to the rocket

(GR8677 #61)

Solution:

In free space → without gravity → no external force:

$\\F= \frac{dp}{dt}=\frac{dmv}{dt}=m\frac{dv}{dt}+v \frac{dm}{dt}=0 \\\\\rightarrow m\frac{dv}{dt}=-v_\text{exhaust}\frac{dm}{dt}$

The equation of motion:

$m \frac{dv}{dt}+u \frac{dm}{dt}=0 \rightarrow m \frac{dv}{dt}= -u \frac{dm}{dt}$

→ u = v_exhaust relative to the rocket.

Answer: E

Physics Problems & Solutions

Classical Mechanics - Equation of Motion

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