# Momentum and Energy problem (ball and incline collision)

• Idoke
In summary, a ball with a mass of 6kg and an unknown initial speed collides with a stationary inclined plane of mass 42kg at an angle of 10 degrees. Using equations for conservation of momentum and energy, it is possible to eliminate one variable and solve for the ratio of the initial and final speeds. This can then be used in the equation for conservation of momentum on the inclined plane's axis to solve for the unknown angle, alpha.
Idoke

## Homework Statement

A ball, with mass $$m_{1}=6 kg$$ and speed $$v_{0}$$ (unknown) hits an inclined plane (as shown) of mass $$m_{2}=42 kg$$ at rest on a frictionless floor.
there is no friction between the ball and the plane.
the angle $$\beta$$ is 10 degrees.
Question: what is the value of $$\alpha$$?2. Diagram

## The Attempt at a Solution

I used three things:
1. The conservation of momentum parallel to the X axis, because there are no forces on the system in that axis:
$$m_{1}v_{0}=m_{1}v_{1}cos\beta+m_{2}v_{2}$$
where $$v_{1}$$ and $$v_{2}$$ are the velocities after the collision of the ball and plane, respectively.
EDITED: the equation was wrong, the inclined plane had no velocity before the collision.

2. The conservation of the ball's momentum on the axis parallel to the inclined plane (because no forces are acting on the ball in that axis) :
$$m_{1}v_{0}sin\alpha=m_{1}v_{1}sin\beta$$

3. Conservation of kinetic energy:
$$1/2m_{1}v_{0}^{2}=1/2m_{1}v_{1}^{2}+1/2m_{2}v_{2}^{2}$$Now I have a problem, I have 3 equations but 4 unknowns (v1, v2, alpha and v0). could someone shed some light on this?
I thought about doing this in the center of mass reference frame but i always get stuck with the fact that the momentum is not conserved in the Y direction.
help would be appreciated.
Thanks.

#### Attachments

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You can eliminate v2 from the equations for the energy and horizontal momentum, and than solve for v1/v0.

ehild

You can eliminate v2 from the equations for the energy and horizontal momentum, and than solve for v1/v0.

But I still need to find alpha!
Do I not have the same predicament?
I feel like I'm missing another equation, or there might be a more elegant solution than this mess.
If I could find how much impulse (or change in momentum) the floor had on the ball and plane I think I could arrive at a solution.
What do you think?

from your second eqn V1/V0=sin (alpha)/sin (beta); the latter is known.

First of all, thanks for your time, both of you.
Now, I fixed the first equation, the inclined plane had no velocity before the collision.
I might be really dense, but I don't understand how to eliminate v2 from my equation or how that helps me solve the system.
If you could clarify, i'd be very grateful.
Ido.

I believe what ehild was suggesting is using eqns 1 & 3 to get the ratio v1/vo

in other words: use eqn 1 to express v2 in terms of masses, angles and v1

something like v2=m1/m2 * (1-cos(alpha)) v1. Then use your conservation of energy formula but substitute for v2 the eqn above. You should be able to get a v1/vo ratio from that. Then use eqn 2 as I mentioned.

## 1. What is momentum and energy in the context of a ball and incline collision?

Momentum is a measure of an object's motion, calculated by multiplying its mass by its velocity. Energy is the ability to do work, and in this context, it refers to the kinetic energy of the ball as it moves along the incline.

## 2. How do you calculate the momentum of a ball after a collision on an incline?

The momentum of the ball after the collision can be calculated using the formula p = mv, where p is the momentum, m is the mass of the ball, and v is its velocity.

## 3. What factors affect the energy of the ball in a collision on an incline?

The energy of the ball in a collision on an incline can be affected by factors such as the mass and velocity of the ball, the angle of the incline, and the presence of any external forces like friction.

## 4. How is the conservation of momentum and energy applied in a ball and incline collision?

The law of conservation of momentum states that the total momentum of a system remains constant before and after a collision. In the case of a ball and incline collision, the initial momentum of the ball will equal the final momentum after the collision. The law of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed. In this case, the initial kinetic energy of the ball will equal the final kinetic energy after the collision.

## 5. What are some real-life applications of understanding momentum and energy in collisions?

Understanding the principles of momentum and energy in collisions is crucial in fields such as physics, engineering, and sports. It is also important in safety measures, such as designing car airbags to reduce the impact force during a collision.

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