Question

A 6090 kgspace probe moving nose-first toward Jupiter at 105 m/srelatives to the Sun fires its rocket engine, ejectingof exhaust at a speed of 253 m/srelatives to the space probe. What is the final velocity of the probe?

Short answer

Final velocity of the probe is 108m/s.

Step-by-step solution

Listing the given quantities

Mass of the space probe is$M_i=6090\mathrm{kg}$

Mass of the exhaust is$M_e=80\mathrm{kg}$

Speed of the exhaust relative to the space probe is,$v_{rel}=253\mathrm{m}/\mathrm{s}$

Initial speed of the space probe is,$v_{i,s}=105\mathrm{m}/\mathrm{s}$

Concept

Using the second rocket equation we can find the final velocity of the probe.

Formula:

Second rocket equation gives,

$v_f-v_i=v_{rel}In\left(\frac{M_i}{M_f}\right)$

Calculation of final velocities

Final mass of the system

$$\begin{gathered} M_i-M_e=6090-80 \\ =6010\mathrm{kg} \end{gathered}$$

Initial speed of the system is,

$$\begin{gathered} v_i=v_{i,s}+v_{i,e} \\ =105\mathrm{m}/\mathrm{s}+0 \\ =105\mathrm{m}/\mathrm{s} \end{gathered}$$

Second rocket equation gives,

$$\begin{gathered} v_f-v_i=v_{rel}In\left(\frac{M_i}{M_f}\right) \\ =105+\left(253\right)\mathrm{In}\left(\frac{6090}{6010}\right) \\ =108.35 \\ ~108\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, Final velocity of the probe is 108 m/s