Hydraulic Piston on te bottom of a hill

[jenha14]

At a ride in an amusement part a cart, with a mass of 150 kg, rolls down a long curved hill. The height of the hill is 50 m and the length of the track that the cart follows is 300 m. As the cart rolls down the hill, it experiences both a rolling frictional force and air resistance. The net effect of these two resistive forces over the run is equivalent to a constant force of 70 N, which is opposite to the cart's velocity.
(A) If the cart starts out at rest at the top of the hill, what is the speed at the bottom?
(B) There is a large hydraulic piston at the bottom of the hill thate engages the cart and brings it to rest by applying a force for 5.0 m. Neglecting all other resitive forces, what is the magnitude of the force needed to stop the cart?

RELEVENT EQUATIONS

(Uf - Ui) + (Kf - Ki) = -Fd

ATTEMPT AT THE SOLUTION

I found (A) to be 26.4 m/s, but I have no idea on where to start on part (B)

 

[Doc Al]

I found (A) to be 26.4 m/s,

Looks good.

but I have no idea on where to start on part (B)

You can solve part (B) in any of several ways. You can use energy methods: How much work is required to bring it to rest? Or you can use kinematics: What acceleration is needed to bring it to rest in the given distance?

 

[ogb p]

Your answer for part (A) is correct. To solve part (B), we can use the relevant equation you provided: (Uf - Ui) + (Kf - Ki) = -Fd.

To start, we need to find the initial kinetic energy (Ki) of the cart at the bottom of the hill. We know that the cart has a mass of 150 kg and a speed of 26.4 m/s. So, Ki = 1/2 * 150 kg * (26.4 m/s)^2 = 110880 J.

Next, we need to find the final potential energy (Uf) of the cart at the bottom of the hill. We know that the height of the hill is 50 m and the mass of the cart is 150 kg. So, Uf = mgh = 150 kg * 9.8 m/s^2 * 50 m = 73500 J.

Now, we can plug in the values into the equation: (Uf - Ui) + (Kf - Ki) = -Fd.

(73500 J - 0 J) + (Kf - 110880 J) = -Fd

Solving for Kf, we get Kf = 184380 J.

This is the final kinetic energy of the cart right before the hydraulic piston engages it. Now, we can use the work-energy theorem to find the force needed to stop the cart.

W = F * d = (Kf - Ki)

Solving for F, we get F = (Kf - Ki)/d.

Plugging in the values, we get F = (184380 J - 110880 J)/5.0 m = 14700 N.

So, the magnitude of the force needed to stop the cart is 14700 N. This is the force that the hydraulic piston needs to apply to bring the cart to a stop in 5.0 m.