Question

The National Transportation Safety Board is testing the crashworthiness of a new car. The \(2340-\mathrm{kg}\) vehicle is driven at \(12.6 \mathrm{~km} / \mathrm{h}\) into an abutment. During impact, the center of mass of the car moves forward \(64.0 \mathrm{~cm} ;\) the abutment is compressed by \(8.30 \mathrm{~cm}\). Ignore friction between the car and the road. (a) Find the force, assumed constant, exerted by the abutment on the car. (b) By how much does the internal energy of the car increase?

Short answer

The force exerted by the abutment on the car and the increase in the internal energy of the car can be calculated following the above steps. The force is found using the relationship between the work done and the distance over which it is exerted. The increase in internal energy is equal to the work done on the car, which is its initial kinetic energy.

Step-by-step solution

Convert Speed

Let's begin by converting the speed of the car from km/h to m/s as the other units in the problem are in terms of meter and second. \n The conversion factor is \(1 km/hr = 0.278 m/s\). So, \(12.6 km/hr = 12.6 * 0.278 m/s = 3.5 m/s\)

Calculate Initial Kinetic Energy

The initial kinetic energy of the vehicle can be found using the formula: \(KE = 1/2 * mass * velocity^2\). For this car, that would be \(KE = 1/2 * 2340 kg * (3.5 m/s)^2\). Calculate this to find the initial kinetic energy of the car.

Find Total Distance Abutment and Car Move

The total distance that the abutment moves and the car moves forward can be found by adding the two given distances. The distances are in cm, but we need them in meter. The conversion factor is \(1 cm = 0.01 m\). Therefore, the total distance is: \(64.0 cm * 0.01 + 8.30 cm * 0.01 = 0.723 m\)

Calculate Force exerted by the Abutment on the Car

Work done on the car is equal to the change in kinetic energy of the car, which we know is zero because the car stops after the impact. Work done is also equal to the force exerted on it times the distance over which it is exerted: \(Work = Force * distance\). From this, we can solve for the force: \(Force = Work / distance\). Since the total work done on the car is zero, the car exerted an equal amount of work in the opposite direction, which can be labeled as negative work done by the car, or essentially, positive work done on the car. This is the kinetic energy of the car we calculated earlier. Substitute the value of the work done on the car and the total distance into the equation to obtain the force.

Calculate Increase in Internal Energy of the Car

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. As we've calculated, the work done on the car is equal to its initial kinetic energy. Therefore, this is also the increase in the internal energy of the car as a result of the impact on the abutment.

Key concepts

Understanding Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. It depends on two primary factors – the mass of the object and its velocity. The mathematical expression for kinetic energy is given by the formula:

• \[ KE = \frac{1}{2} m v^2 \]

where \( KE \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) represents the velocity.
The unit of measurement for kinetic energy is Joules (\( J \)). In the context of the exercise, we calculated the initial kinetic energy of the car using its mass \((2340 \text{ kg})\) and its converted speed \((3.5 \text{ m/s})\). This energy tells us how much energy is needed to bring the car to its current state of motion.

The Role of Force Calculation

Force calculation is crucial for understanding the interactions between objects during a collision. In physics, force is defined as an interaction that changes the motion of an object. In this exercise, we calculated the force exerted by the abutment on the car during impact.

• Force can be expressed by the equation:\[ F = \frac{W}{d} \]

where \( F \) is the force, \( W \) is the work done, and \( d \) is the distance over which the force is applied.
We determined the total distance by summing the distances moved by both the car and the abutment in meters. This method provided insight into the extent of the collision's impact over the given distance. Force plays a pivotal role in crashworthiness by indicating how much resistance an object can provide or withstand during impact.

The Work-Energy Theorem Explained

The work-energy theorem is a fundamental principle in physics that describes the relationship between work and energy. It states that the work done on an object equals the change in its kinetic energy. This can be summed up in the equation:

• \[ W = \Delta KE \]

where \( W \) represents the work done, and \( \Delta KE \) symbolizes the change in kinetic energy.
In our exercise, this theorem was pivotal as it linked the work done during the car's impact to its initial kinetic energy. Since the car comes to rest after the impact, the final kinetic energy is zero. Therefore, the amount of work done equals the initial kinetic energy, which also represents the change in the car's internal energy during the collision.

Understanding Internal Energy Increase

During a collision, a portion of kinetic energy is transformed into internal energy due to deformation and heat. The internal energy increase is essentially the energy absorbed by the car's structure during the impact.
In the problem, we figured out this increase using the work-energy theorem. Since the initial kinetic energy of the car was the only energy contributing to the process, it converted entirely into internal energy when the car came to a standstill.

• In simpler terms, this internal energy helps to "absorb" the impact, preventing all of it from being transferred to the passengers or other components.

Understanding this concept is vital in crashworthiness design, as it can lead to enhanced safety measures using material deformation and energy distribution techniques to minimize passenger injuries.