# Elastic Collision in Two Reference Frames

• Bonnie
In summary, the problem involves two skateboarders colliding elastically on a ramp, with given masses and starting height. The goal is to calculate the maximum heights each skateboarder reaches after the collision using the conservation of energy and momentum equations. In the laboratory frame, the velocity and momentum of each skateboarder before the collision can be calculated, but the lack of final values after the collision makes it difficult to proceed. However, in the center of momentum frame, the velocities of the skateboarders after the collision can be obtained by using the conservation of momentum equation and the relative velocity between them. From there, the maximum height can be calculated using the formula h = v^2/(2g).
Bonnie

## Homework Statement

1. Two skateboarders start from rest on opposite sides of a ramp like the one in the image, roll down and collide elastically on the level part of the ramp. The masses of the skateboarders are m1 = 48 kg and m2 = 55 kg and they both start from the height h = 4.70m. Ignoring any friction, calculate the maximum heights that each skateboarder reaches after the collision
(a) by performing the collision calculation in the laboratory frame
(b) by performing the collision calculation in the centre of momentum frame

## Homework Equations

Conservation of energy/momentum
1/2mv2 = mgh
mvi = mvf

## The Attempt at a Solution

I worked out the velocity and momentum of each skater at the instant before they collide, thinking that I could then use normal conservation of momentum and energy to work out the same values for afterward, but as I reached this point I realized that I do not know how to proceed because I have no final values to substitute into the post-collision equation, so do not know how to proceed from here. In the lab frame I have calculated the pre-collision momentum of skater 1 to be 460.8kgm/s, and skater 2 to be 528kgm/s

#### Attachments

• skate%20ramp3_001.jpg
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Please show what you have done.

Velocity of both skateboarders on collison will be √(2gh). Both experience the same change in momentum Δp = 2μΔv (elastic collision) where μ is their reduced mass [m1*m2/(m1+m2)] and Δv is their relative velocity = 2√(2gh). Simply subtract Δp from the respective momentums and then divide by respective masses to obtain respective velocities. Finally use h = v^2/(2g).

## 1. What is an elastic collision?

An elastic collision is a type of collision between two objects in which there is no net loss of kinetic energy. This means that the total kinetic energy of the objects before and after the collision remains the same.

## 2. What are the two reference frames involved in an elastic collision?

The two reference frames involved in an elastic collision are the initial reference frame, when the objects are approaching each other, and the final reference frame, when the objects have collided and are moving away from each other.

## 3. How is momentum conserved in an elastic collision?

In an elastic collision, the total momentum of the objects before and after the collision remains the same. This is because the forces acting on the objects during the collision are equal and opposite, resulting in the conservation of momentum.

## 4. What is the difference between elastic and inelastic collisions?

In an elastic collision, there is no net loss of kinetic energy, while in an inelastic collision, some of the kinetic energy is converted into other forms of energy, such as heat or sound. Inelastic collisions also typically result in a change in the objects' velocities, while elastic collisions do not.

## 5. How can the coefficient of restitution be used to determine whether a collision is elastic or inelastic?

The coefficient of restitution is a measure of how much kinetic energy is conserved in a collision. A value of 1 indicates an elastic collision, while a value less than 1 indicates an inelastic collision. Therefore, by calculating the coefficient of restitution, it is possible to determine whether a collision is elastic or inelastic.

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