# Solving the Impulse-Momentum Equation for a Coasting Car

In summary, a problem is given regarding a coasting car on a slope with a mass of 1000 kg and a speed of 100 km/h. The car is slowed down to 50 km/h in 8 seconds when the brakes are applied. Using the impulse-momentum equation, the average braking force exerted by the road on all the tires during this time period is calculated to be 3400 N. The process involves using the net force and frictional force equations and finding the average acceleration of the car. The concept of integrating the equation is questioned and the term "impulse-momentum equation" is discussed.

## 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.

## 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?

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.