Question

A car of mass \(1214.5 \mathrm{~kg}\) is moving at a speed of \(62.5 \mathrm{mph}\) when it misses a curve in the road and hits a bridge piling. If the car comes to rest in \(0.236 \mathrm{~s}\), how much average power (in watts) is expended in this interval?

Short answer

Answer: The average power expended by the car during this interval is approximately 2.198 × 10^6 W.

Step-by-step solution

Convert speed from mph to m/s

First, we need to convert the speed of the car from miles per hour (mph) to meters per second (m/s). To do this, we can use the following conversion factor: 1 mph = 0.44704 m/s So, the speed in m/s is: \(62.5~\text{mph} \times 0.44704~\frac{\text{m}}{\text{s}} = 27.940~\frac{\text{m}}{\text{s}}\)

Calculate the kinetic energy

Now, we can find the kinetic energy of the car. The formula to calculate kinetic energy is: \(KE = \frac{1}{2}mv^2\) Where \(m\) is the mass of the car and \(v\) is its speed. Plugging in the values, we get: \(KE = \frac{1}{2} \times 1214.5~\text{kg} \times (27.940~\frac{\text{m}}{\text{s}})^2 = 518518.5326~\text{J}\) (Joules)

Calculate the work done

As the car comes to rest, its final kinetic energy becomes zero. So, the work done on the car will be equal to the initial kinetic energy (since work-energy theorem states that the work done on an object is equal to the change in its kinetic energy): \(W = KE_\text{initial} - KE_\text{final} = 518518.5326~\text{J} - 0~\text{J} = 518518.5326~\text{J}\)

Calculate the average power

Finally, we can calculate the average power expended by the car as it comes to a stop by dividing the work done by the time taken: \(P_\text{average} = \frac{W}{t} = \frac{518518.5326~\text{J}}{0.236~\text{s}} = 2197925.134~\text{W}\) Therefore, the average power expended by the car during this interval is approximately \(2197925.134~\text{W}\) or \(2.198 \times 10^6 ~\text{W}\).