Particle collision at an angle

[Saxby]

Homework Statement

A particle of mass m traveling at a velocity u makes a perfect elastic collision with a stationary particle. After the collision both particles are observed to be traveling in directions making angles of 30 degrees to the original path of the first particle.

a) Use the laws of the conservation of energy and momentum to write down three equations relating mass and velocities of the particles involved in the collision described above.

b) Solve the equations to find the mass of the seocnd particle and the final velocities of the two particles.


2. Relevent equations
Kinetic energy: Ek = (1/2)*m*v²
Conservation of momentum: (m₁*u₁) + (m₂*u₂) = (m₁*v₁) + (m₂*v₂)
m₁ = Mass of particle that was intially moving
m₂ = Mass of particle that was intially stationary
u₁ = Intial velocity of m₁
v₁ = Final velocity of m₁
v₂ = Final velocity of m₂

3. The attempt at solution
I believe the first equation in the problem is the conservation of momentum, which for this problem i have written as:

m₁*u₁ = [m₁* ((v₁sinθ)² + (v₁cosθ)²)^1/2] + [m₂ * ( (v₂sinθ)² + (v₂sinθ)²)^1/2]

I believe the second equation in the problem is kinetic energy:

(1/2)*m₁*u₁² = (1/2)*m₁*(v_1sinθ² + v_1cosθ²) + (1/2)*m₂*(v_2sinθ² + v_2cosθ²)

For the third equation i have no idea, i don't believe it's rotational energy or anything like that. I think it may have somthing to do with the angles but frankly i don't know. Any help would be much apprietiated :)

 

[Doc Al]

I believe the first equation in the problem is the conservation of momentum, which for this problem i have written as:

m₁*u₁ = [m₁* ((v₁sinθ)² + (v₁cosθ)²)^1/2] + [m₂ * ( (v₂sinθ)² + (v₂sinθ)²)^1/2]

Momentum is a vector so you have to treat is as such. Write separate momentum conservation equations for components parallel and perpendicular to the original direction. (That's how you'll end up with three equations.)

 

[Saxby]

Momentum is a vector so you have to treat is as such. Write separate momentum conservation equations for components parallel and perpendicular to the original direction. (That's how you'll end up with three equations.)

Thank you, that makes sense. But how do i know what percent of the original momentum goes in the y-direction of both particles and how much goes in the x-direction?

 

[Doc Al]

Thank you, that makes sense. But how do i know what percent of the original momentum goes in the y-direction of both particles and how much goes in the x-direction?

You don't need to know anything. Just set up an equation for the x-components and another for the y-components. You have the angles. I would choose the original direction to be along the +x axis.

 

[Nickie]

I can help you with solving this problem. Firstly, you are correct in identifying the first two equations as the conservation of momentum and kinetic energy equations. However, in the first equation, the final velocities of both particles should be written as (v1cos30) and (v2cos30) respectively, as the angle of 30 degrees is given in the problem statement.

For the third equation, we can use the fact that the collision is perfectly elastic, meaning that the total kinetic energy of the system before and after the collision remains the same. Therefore, we can write the equation as follows:

(1/2)*m1*u1^2 = (1/2)*m1*(v1cos30)^2 + (1/2)*m2*(v2cos30)^2

Now, we have three equations with three unknowns (m2, v1, and v2), and we can solve them simultaneously to find the values of the second particle's mass and the final velocities of both particles. I would recommend using substitution or elimination methods to solve the equations.

I hope this helps and good luck with your homework! Remember to always check your units and make sure they are consistent throughout the equations.