What Determines the Maximum Compression of a Spring in an Elastic Collision?

[phizics09]

Homework Statement

A 1 kg cart is at the top of a frictionless ramp as shown. It rolls down and makes an elastic collision with a stationary 3 kg cart at the bottom. The spring constant is 400 N/m. What is the maximum compression of the spring? How far up the ramp will the first cart travel after the collision?

The Attempt at a Solution

I got part a, but I don't know how to do part b.

Thanks!

 

[SammyS]

Homework Statement

A 1 kg cart is at the top of a frictionless ramp as shown. It rolls down and makes an elastic collision with a stationary 3 kg cart at the bottom. The spring constant is 400 N/m. What is the maximum compression of the spring? How far up the ramp will the first cart travel after the collision?

The Attempt at a Solution

I got part a, but I don't know how to do part b.

Thanks!

No figure.

Not enough detail given -- especially without the figure !

 

[phizics09]

sorry i forgot to attach the picture !

 

[SammyS]

How did you find the maximum compression of the spring? --- in detail.

 

[asteriatic]

I would approach this problem by first analyzing the given information and identifying any relevant equations or principles that can be applied. From the given information, we know that there is a 1 kg cart at the top of a frictionless ramp, which will roll down and collide with a stationary 3 kg cart at the bottom. The spring constant is also given as 400 N/m.

To find the maximum compression of the spring, we can use the conservation of energy principle. At the top of the ramp, the cart has potential energy given by mgh, where m is the mass of the cart, g is the acceleration due to gravity, and h is the height of the ramp. As it rolls down the ramp, this potential energy is converted into kinetic energy, given by 1/2mv^2, where v is the velocity of the cart at the bottom of the ramp.

At the bottom of the ramp, the cart collides with the stationary 3 kg cart. Since the collision is elastic, we can use the conservation of momentum principle to find the final velocity of the carts after the collision. This velocity can then be used to calculate the compression of the spring using Hooke's Law, F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.

To find how far up the ramp the first cart will travel after the collision, we can use the conservation of energy principle again. At the bottom of the ramp, the kinetic energy of the carts will be converted back into potential energy, given by mgh, where m is the combined mass of the carts and h is the height the carts reach after the collision. We can then solve for h to find the distance traveled up the ramp.

In conclusion, as a scientist, I would approach this problem by using the principles of conservation of energy and momentum, as well as relevant equations such as Hooke's Law, to find the maximum compression of the spring and the distance traveled up the ramp after the collision.