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: To find the average power expended, first, convert the speed from mph to m/s, calculate the initial and final kinetic energy of the car, and use the work-energy theorem to find the work done. Then, divide the work done by the time taken for the car to come to rest, which is 0.236 seconds.

Step-by-step solution

Convert the speed to SI units (meters per second)

First, we need to convert the speed from miles per hour (mph) to meters per second (m/s). The conversion factor is 1 mph = 0.44704 m/s. So, \(62.5\) mph \(= 62.5 * 0.44704\) m/s

Calculate the initial kinetic energy of the car

To find the initial kinetic energy (KE) of the car, we can use the following equation: KE = \(\frac{1}{2} * m * v^2\) where m is the mass of the car and v is its speed. In this case, m = \(1214.5\) kg and v = \(62.5 * 0.44704\) m/s.

Calculate the final kinetic energy of the car

Since the car comes to rest after hitting the bridge piling, its final speed is 0 m/s. Therefore, the final kinetic energy of the car is 0 J.

Calculate the work done using the work-energy theorem

Now, we can use the work-energy theorem to determine the work done in bringing the car to rest. The theorem states: Work done (W) = Change in kinetic energy (ΔKE) ΔKE = Final KE - Initial KE

Calculate the average power

Finally, we can find the average power (P) expended during the process by dividing the work done (W) by the time interval (t) in which the car comes to rest. P = \(\frac{W}{t}\), where t = \(0.236\) s.

Key concepts

Kinetic Energy

Imagine you're pushing a toy car across the floor. The effort you put in to get it moving is similar to what happens when real cars glide down the road. This energy is what we call kinetic energy, the energy of motion.

Kinetic energy (KE) is calculated using the equation: \[ KE = \frac{1}{2} \times m \times v^2 \]where m represents the mass of an object, and v is the velocity (or speed) at which it's moving. In the context of our textbook exercise, a car with a mass of 1214.5 kg traveling at 62.5 mph possesses kinetic energy, which can be found by converting its speed to meters per second and plugging the values into the equation.

Work-Energy Theorem

Have you ever tried stopping a moving skateboard with your foot? You exert force, and thus do work, to bring it to a halt. This is where the work-energy theorem comes into play.

The work-energy theorem articulates a fundamental concept: the work done on an object is equal to the change in its kinetic energy, expressed as:\[ \text{Work done} = \Delta KE = \text{Final KE} - \text{Initial KE} \]In our car scenario, when the vehicle crashes and ceases to move, it has gone from a state of kinetic energy to one of zero kinetic energy. According to the theorem, this difference in kinetic energy equals the work done on the car by the forces that stopped it. By calculating this work done, we dive deeper into understanding how energy transforms during the event of a collision.

Unit Conversion

Picture this—a recipe calls for 3 teaspoons, but all you have is a tablespoon measurement. This is when you need a unit conversion—turning one type of unit into another. In science, especially physics, converting units correctly is vital to getting the right answers.

In our original problem, we had to convert the car speed from miles per hour (mph) to meters per second (m/s), a common type of unit conversion in kinematics. Accuracy in these conversions ensures that all subsequent calculations, such as kinetic energy and power, are also accurate. By using the conversion factor (1 mph = 0.44704 m/s), we can seamlessly switch from one system of measurement to another, allowing for proper application of physics equations which typically use the International System of Units (SI).