Question

Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0m/s. The total mass is initially 75.0 kgand is 70kgafter the extinguisher is fired.

Short answer

The exhausted velocity is 145 m/s .

Step-by-step solution

Definition of Velocity

It is the speed at which the object's position in relation to the reference frame and time changes.

Expression to find the final velocity

The equation to find the final velocity of the rocket is

\[{\rm{v = }}{{\rm{v}}_{\rm{0}}}{\rm{ + }}{{\rm{v}}_{\rm{e}}}{\rm{ln}}\left( {\frac{{{{\rm{m}}_{\rm{o}}}}}{{{{\rm{m}}_{\rm{r}}}}}} \right)\]

Where m_o is the initial mass, m_r is the remaining mass, v_o is the initial velocity and v is final velocity

Calculation

Substituting 10.0 m/s for v, 0 m/s for v_o, 75 kg for m_o, and 70.0 kg for m_r

\begin{align} {{\rm{10}}{\rm{.0m/s = 0 m/s + }}{{\rm{v}}_{\rm{e}}}{\rm{ln}}\left( {\frac{{{\rm{75}}{\rm{.0 kg}}}}{{{\rm{70}}{\rm{.00 kg}}}}} \right)}\end{align} \begin{align} {{\rm{10}}{\rm{.0m/s = }}{{\rm{v}}_{\rm{e}}}{\rm{(0}}{\rm{.0689)}}}\end{align} \begin{align} {{\rm{v}}_{\rm{e}}}{\rm{ = }}\frac{{{\rm{10}}{\rm{.0 m/s}}}}{{{\rm{0}}{\rm{.0689}}}}\end{align} \begin{align} {{\rm{v}}_{\rm{e}}}{\rm{ = 145 m/s}}\end{align}

Hence the velocity exhausted is 145m/s.