A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest.

Solution:

Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. In symbols, Newton’s second law of motion is defined to be

\textbf{F}_{\text{net}} = \frac{\Delta \textbf{p}}{\Delta t} ,

where \textbf{F}_{\text{net}} is the net external force, \Delta \textbf{p} is the change in momentum, and \Delta t is the change in time.

For this problem, we are given the following values:

\begin{align*}
m & = 15000\ \text{kg} \\
\textbf{v}_{\text{initial}} & = 5.4\ \text{m}/\text{s} \\
\textbf{v}_{\text{final}} & = 0\ \text{m}/\text{s} \\
\textbf{F}_{\text{net}} & = 1500\ \text{N}
\end{align*}

Substitute these given values in the equation above.

\begin{align*}
\textbf{F}_{\text{net}} & = \frac{\Delta \textbf{p}}{\Delta t} \\
\Delta t  & = \frac{\Delta \textbf{p}}{\textbf{F}_{\text{net}}} \\
\Delta t  & = \frac{m \left( \Delta\textbf{v} \right)}{\textbf{F}_{\text{net}}} \\
\Delta t  & = \frac{15000\ \text{kg} \left( 5.4\ \text{m}/\text{s} \right)}{1500\ \text{N}} \\
\Delta t  & = 54\ s \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

It would take 54 seconds to stop the car.