Solving the Impulse-Momentum Equation for a Coasting Car

[madahmad1]

Homework Statement

A coasting car with a mass of 1000 kg has a speed of 100 km/h down a 10
degree slope when the brakes are applied. if the car is slowed to a speed of 50
km/ h in 8s, compute using the impulse-momentum equation the average of the total braking force exerted by the road on all the tires during the period. Treat the car as a particle and neglect air resistance.

Homework Equations

The Attempt at a Solution

The impulse-momentum equation states that the sum of Forces F = mdv/dt
so we turn this equation into an integral and solve, I know that the answer is 3400 N but do not know how to get it. any help?

 

[misho]

For this, I'm going to take up the ramp to be the positive direction and down the ramp to be the negative direction.

Let F be the net force (this should end up being positive, since the initial velocity is negative, based on the definition above, and the car is slowing down).
Let f be the frictional force (should be positive, for the same reasons).

By drawing the FBD, you should see that:

$$F = f - mg \sin 10 ^\circ$$
OR:
$$f = F + mg \sin 10 ^\circ$$

Also, we know the average acceleration of the car, so we know F:

$$F = m \frac{dv}{dt} = ma = m \frac{ (- \frac{50}{3.6})-(-\frac{100}{3.6}) }{ 8 } = m \frac{50}{8*3.6}$$

The rest is pretty obvious (I got 3439 N, which I figure is close enough).

I'm not sure why you would want to integrate F = mdv/dt, since you're looking for a force.

Just out of curiosity, which form of Newton's 2nd law were you taught is called the "impulse-momentum equation"? It's the first time I've heard the term used.

 

[piercas]

I would approach this problem by first understanding the physical principles involved. The impulse-momentum equation is a fundamental equation in mechanics that relates the change in momentum of an object to the net force exerted on it. In this case, the coasting car experiences a change in momentum as it slows down due to the application of the brakes.

To solve for the average braking force, we can use the impulse-momentum equation in its integral form:

∫ F dt = m∫ dv

Where F is the force, t is time, m is the mass of the car, and v is the velocity.

We know that the car has a mass of 1000 kg and is initially travelling at a speed of 100 km/h, which is equivalent to 27.78 m/s. The car is then slowed down to a speed of 50 km/h or 13.89 m/s in 8 seconds.

Using the integral form of the impulse-momentum equation, we can solve for the average braking force:

∫ F dt = m∫ dv
F∫ dt = mvf - mvi
F(8s) = (1000 kg)(13.89 m/s) - (1000 kg)(27.78 m/s)
F = (13890 - 27780) kgm/s^2
F = -13890 kgm/s^2
F = -13890 N

However, since the question asks for the average of the total braking force exerted by all four tires, we need to divide this value by 4. This gives us an average braking force of 3472.5 N for each tire, which rounds to 3400 N.

In conclusion, using the impulse-momentum equation, we can calculate the average braking force exerted by each tire on the coasting car to be 3400 N. This shows that the car experienced a significant force in order to slow down from 100 km/h to 50 km/h in just 8 seconds.