Finding kinetic energy after an elastic collision

[Delta Sheets]

Homework Statement

A ball with mass m = 0.230 kg and kinetic energy K1 = 1.16 J collides elastically with a second ball of thesame mass that is initially at rest. After the collision, the first ball moves away at an angle of θ1= 37.6° with respect to the horizontal. What is the kinetic energy of the first ball after the collision?

Homework Equations

KE=1/2mv^2
m1v1 + m2v2(initial)=m1v1 + m2v2(final)

The Attempt at a Solution

Since kinetic energy is conserved,I figured the kinetic energy of the two balls would be the same after the collision at the θ=0 position. So I then tried to solve for kinetic energy in the direction of θ=37.6. Seeing as the two kinetic energies of the masses should equal 1.16 J, I am lost on the problem and do not know where to start. I would like to be put in the right direction, not necessarily given an answer so I can figure the work out on my own.

 

[SammyS]

Homework Statement

A ball with mass m = 0.230 kg and kinetic energy K1 = 1.16 J collides elastically with a second ball of the same mass that is initially at rest. After the collision, the first ball moves away at an angle of θ1= 37.6° with respect to the horizontal. What is the kinetic energy of the first ball after the collision?

Homework Equations

KE=1/2mv^2
m1v1 + m2v2(initial)=m1v1 + m2v2(final)

The Attempt at a Solution

Since kinetic energy is conserved,I figured the kinetic energy of the two balls would be the same after the collision at the θ=0 position.

That's not a valid assumption.

So I then tried to solve for kinetic energy in the direction of θ=37.6. Seeing as the two kinetic energies of the masses should equal 1.16 J, I am lost on the problem and do not know where to start. I would like to be put in the right direction, not necessarily given an answer so I can figure the work out on my own.

If the collision were such that there was no deflection to either side, then after the collision, the first ball would be at rest and the second ball would move with the same velocity that the first ball initially had.

 

[Delta Sheets]

That's not a valid assumption.

If the collision were such that there was no deflection to either side, then after the collision, the first ball would be at rest and the second ball would move with the same velocity that the first ball initially had.

Now I understand that part, but in what way can i calculate the kinetic energy of each?

 

[SammyS]

Now I understand that part, but in what way can i calculate the kinetic energy of each?

The conservation of momentum equation is a vector equation.

Therefore,

$\displaystyle \left(\,m_1\,(v_1)_x\ +\ m_2\,(v_2)_x\,\right)_\text{initial} =\left(\,m_1\,(v_1)_x\ +\ m_2\,(v_2)_x\,\right)_\text{final}$

$\displaystyle \left(\,m_1\,(v_1)_y\ +\ m_2\,(v_2)_y\,\right)_\text{initial} =\left(\,m_1\,(v_1)_y\ +\ m_2\,(v_2)_y\,\right)_\text{final}$​

Assuming that the initial velocity is in the x direction, what does that tell you about how the y-components of the final velocities of the two balls compare?

 

[Delta Sheets]

The conservation of momentum equation is a vector equation.

Therefore,

$\displaystyle \left(\,m_1\,(v_1)_x\ +\ m_2\,(v_2)_x\,\right)_\text{initial} =\left(\,m_1\,(v_1)_x\ +\ m_2\,(v_2)_x\,\right)_\text{final}$

$\displaystyle \left(\,m_1\,(v_1)_y\ +\ m_2\,(v_2)_y\,\right)_\text{initial} =\left(\,m_1\,(v_1)_y\ +\ m_2\,(v_2)_y\,\right)_\text{final}$​

Assuming that the initial velocity is in the x direction, what does that tell you about how the y-components of the final velocities of the two balls compare?

This shows the y components of velocity should equal 0 when added together

 

[SammyS]

This shows the y components of velocity should equal 0 when added together

Right.

For the first ball, you know how the x-component final velocity is related to the y-component final velocity.

 

[Delta Sheets]

Right.

For the first ball, you know how the x-component final velocity is related to the y-component final velocity.

The problem was much easier than thought. Find velocity of the first object from kinetic energy. Use this velocity times the cosine of the angle. Use that velocity for finding the new kinetic energy. Giving an answer of 0.728J.