Problem 8 - University Physics with Modern Physics

Question

Which of the following statements about car collisions are true and which are false? a) The essential safety benefit of crumple zones (parts of the front of a car designed to receive maximum deformation during a head-on collision) results from absorbing kinetic energy, converting it into deformation, and lengthening the effective collision time, thus reducing the average force experienced by the driver. b) If car 1 has mass \(m\) and speed \(v\), and car 2 has mass \(0.5 m\) and speed \(1.5 v\), then both cars have the same momentum. c) If two identical cars with identical speeds collide head on, the magnitude of the impulse received by each car and each driver is the same as if one car at the same speed had collided head on with a concrete wall. d) Car 1 has mass \(m,\) and car 2 has mass \(2 m .\) In a head-on collision of these cars while moving at identical speeds in opposite directions, car 1 experiences a bigger acceleration than car 2 . e) Car 1 has mass \(m\), and car 2 has mass \(2 m\). In a headon collision of these cars while moving at identical speeds in opposite directions, car 1 receives an impulse of bigger magnitude than that received by car 2.

Step-by-Step Solution

Answer

Explain your answer.

Step 1: Statement a

By analyzing the statement, we can see that it refers to crumple zones and their impact on car collisions. The principle being discussed here is the conservation of energy, specifically the conversion of kinetic energy into deformation energy. As the crumple zone is designed to deform during a collision, it helps to absorb kinetic energy by transforming it into deformation energy. This results in longer collision time and reduced average force on the driver. This statement is true.

Step 2: Statement b

In this statement, we need to calculate the momentum of the two cars and compare them to see if they have the same momentum. The momentum (p) of an object is given by its mass times its velocity, that is, \(p=mv\). For car 1, its momentum is \(mv\), while for car 2, its momentum is \((0.5m)(1.5v) = 0.75mv\). As we see, the momenta are not equal, and thus the statement is false.

Step 3: Statement c

The statement refers to the impulse received by a car during a collision. The impulse (J) is given by the change in momentum, and in the case of a head-on collision between two identical cars, it's the same as if one car collided with a concrete wall. Since the cars have identical masses and speeds, their momenta before the collision are equal and opposite. After the collision, their momenta would be zero on both cars, as well as on a car that collides head-on with a concrete wall. Hence, the statement is true.

Step 4: Statement d

The statement discusses acceleration during a head-on collision. The acceleration experienced by either car can be found by using Newton's second law, \(F=ma\). Since they have equal and opposite velocities, and the net force acting on them is equal, we can say that car 1 will experience a greater acceleration due to its lesser mass. So, this statement is true.

Step 5: Statement e

This statement analyzes the impulse experienced by the two cars during a head-on collision. Impulse, the product of force and time, is also equal to the change in momentum. As we saw in the previous step, the net force acting on them is equal. Therefore, since they both experience the same amount of force, the time over which this force acts, and thus the impulse received during the collision, should also be the same for both cars. This statement is false.

Key Concepts

Conservation of Energy

Understanding the conservation of energy is fundamental when analyzing car collisions. When two cars collide, the kinetic energy of the vehicles just before the collision is converted into other forms of energy, such as heat, sound, and the energy required to deform the cars. Crumple zones play a vital role in this energy transformation. They are meticulously engineered to crumple in a controlled fashion, absorbing energy and increasing the collision duration.

As a result, the average force exerted on the passengers and drivers is significantly reduced, which is critical in minimizing injuries in an accident. This principle directly correlates with the adage 'it's not the fall that kills you, but the sudden stop at the end.' By lengthening the time over which the stop occurs, the crumple zone spreads the force over a longer period, thereby following the rule that energy must be conserved—transferring kinetic into deformation energy and not disappearing.

Momentum

Momentum is another crucial concept in the physics of car collisions. It is a measure of an object's motion, which is the product of its mass and velocity. Under the principle of conservation of momentum, the total momentum of a system remains constant if no external forces act on it.

In the context of car collisions, when two vehicles crash, the momentum just before the impact must equal the total momentum after the impact, assuming no external forces, like friction or external interventions, significantly affect it. This conservation law helps us predict the outcomes of collisions, such as how vehicles will move post-collision or the forces involved. Misunderstanding momentum, as in the incorrect statement regarding two cars having the same momentum because of different masses and velocities, can lead to a fundamental misinterpretation of the collision dynamics.

Impulse

The impulse delivered during a collision is another aspect illuminated by physics principles. Impulse is the change in momentum of an object and can be calculated as the force applied times the time over which it is applied. It is intimately related to the force experienced by the occupants of a car during a crash.

Equal and Opposite Impulse

When identical cars collide head-on at the same speed, each car experiences an impulse equal in magnitude but opposite in direction to the other's. This is because they exert equal and opposite forces on each other for the same amount of time during the collision, a direct application of Newton's third law of motion. This equality in impulse helps in understanding the effects on each vehicle and in designing safety features for cars.

Newton's Second Law

At the heart of car collision analysis is Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\(F = ma\)). This law helps to understand the acceleration vehicles undergo when forces are applied during a collision.

From this law, we infer that if two cars of different masses collide and the force experienced by each is the same, the car with the lesser mass will have a greater acceleration. In real-world scenarios, this means that lighter vehicles generally sustain greater changes in velocity in a collision, often resulting in more significant damage and higher risks to the occupants. Newton's second law is essential for interpreting the effects of collisions on vehicles of different masses and the resulting accelerations, providing insights for vehicle safety designs and regulations.