A car traveling at a speed of 18 m/s (approximately 40 MPH) crashes into a solid concrete wall. The driver has a mass of 90 kg.
a. What is the change in momentum of the driver as he comes to a stop:
b. What impulse is required in order to produce this change in momentum:
c. How does the application and magnitude of this force differ in two cases: the first, in which the driver is wearing a seat belt, and the second, in which he is not wearing a seat belt and is stopped instead by contact with the windshield and steering column? Will the time of action of the stopping force change? Explain
a. -1600kg*m/s
b. -1600000 N
a. The change in momentum of the driver can be calculated using the formula: Δp = m * Δv, where Δp is the change in momentum, m is the mass of the driver, and Δv is the change in velocity. Since the driver comes to a stop, Δv would be equal to the initial velocity (18 m/s) multiplied by -1. So, Δp = 90 kg * (-18 m/s) = -1620 kg*m/s.
b. The impulse required to produce this change in momentum can be calculated using the formula: J = Δp, where J is the impulse and Δp is the change in momentum. In this case, the impulse required would be equal to the change in momentum, so J = -1620 kg*m/s.
c. When the driver is wearing a seat belt, the force required to stop the driver will be distributed over a longer time period. This is because the seat belt allows for a longer deceleration time due to its elasticity. As a result, the magnitude of the force experienced by the driver will be lower, but the time of action of the stopping force will be longer.
On the other hand, when the driver is not wearing a seat belt, the force required to stop the driver will be concentrated over a much shorter time period. The windshield and the steering column will exert a higher force on the driver, resulting in a greater magnitude of the stopping force. As a result, the time of action of the stopping force will be shorter.
Remember, it is always important to wear a seat belt while driving to minimize the risk of injury in case of an accident. Safety first, folks!
a. To find the change in momentum (Δp) of the driver, we can use the formula:
Δp = m * Δv
Where m is the mass of the driver and Δv is the change in velocity.
Given that the mass of the driver is 90 kg and the car comes to a stop, the change in velocity is equal to the initial velocity of the car (18 m/s).
Δp = 90 kg * 18 m/s
= 1620 kg.m/s
Therefore, the change in momentum of the driver is 1620 kg.m/s.
b. Impulse (J) can be calculated using the formula:
J = Δp
Given that Δp is 1620 kg.m/s as calculated above, the impulse required to produce this change in momentum is 1620 kg.m/s.
c. The force applied to stop the driver will be different in the two cases (wearing a seat belt or not wearing a seat belt).
In the first case, when the driver is wearing a seat belt, the force will be distributed over a larger area as the seat belt applies an equal and opposite force over a larger area on the driver's body. This helps to reduce the magnitude of the force experienced by the driver.
In the second case, when the driver is not wearing a seat belt and is stopped by contact with the windshield and steering column, the force is concentrated on a smaller area of the body, potentially causing more severe injuries.
The magnitude of the force will depend on the time taken to stop the driver. In the second case, where the driver is not wearing a seat belt, the time of action of the stopping force will be shorter compared to the first case, where the driver is wearing a seat belt. This shorter time of action can result in higher peak forces being exerted on the driver's body, potentially causing more severe injuries.
To answer these questions, we need to use the principles of momentum and impulse.
a. The change in momentum of an object is equal to the final momentum minus the initial momentum. In this case, since the driver goes from a positive momentum to zero momentum when coming to a stop, the change in momentum is equal to the negative of the initial momentum.
The initial momentum of the driver can be calculated using the formula: Initial momentum = mass × velocity. Therefore, the initial momentum of the driver is 90 kg × 18 m/s = 1620 kg·m/s.
So, the change in momentum of the driver when coming to a stop is -1620 kg·m/s.
b. Impulse is defined as the change in momentum of an object. In this case, the impulse required to produce the change in momentum can be calculated by multiplying the change in momentum by the respective change in time.
Since the driver comes to a stop, the change in time is the time it takes for the car to come to a full stop upon impact.
Unfortunately, without knowing the specific time it takes for the car to come to a stop, we can't calculate the impulse directly. We need to know the deceleration of the car or the time it takes to stop to determine the impulse.
c. The application and magnitude of the force differ in the two cases: wearing a seat belt or not.
In the case where the driver is wearing a seat belt, the stopping force is distributed over a larger area of the body, typically the chest and hips. The seat belt extends the time it takes for the driver's momentum to change, resulting in a smaller force acting on the body.
In the case where the driver is not wearing a seat belt, the force is applied directly to specific areas of the body that come into contact with the windshield and steering column. This more concentrated force can cause more severe injuries.
The magnitude of the force depends on the time it takes to stop the driver. With a seat belt, the stopping force is spread out over a longer period, resulting in a less forceful impact. Without a seat belt, the time of action of the stopping force is reduced, causing a more sudden and intense force.
Overall, wearing a seat belt increases the time it takes for the change in momentum to occur, reducing the force experienced by the driver and potentially reducing the risk of injury.