Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in a glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed vi. After the collision, the orange disk moves along a direction that

makes an angle theta with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine:
1) the final speed of each disk;
2) whether the collision was elastic or not.

I've seen this question many times on the internet and am bothered with the erroneous answers that were provided. I know that you are most likely done with this class (and hopefully you have passed it) but for the sake of anyone else who is looking for an answer here is the solution:

Common versions of this question have given theta values of 37 degrees for the orange shuffleboard and 53 degrees for the green shuffleboard.

M(green)=M(orange)

M(green)
v(initial) = 0
v(final)=?

M(orange)
V(initial) = 5 m/s (along the horizontal)
V(final) = ?

This is an elastic collision, use the equation for this collision type (MV1 + MV2)i = (MV1-MV2)f (the M's cancel out since both masses are equal)

This is a 2-D collision so split them accordingly:
X: 5 = V1(cos53degrees) + V2(cos37degrees)

Y: 0 = V1(sin37degrees) - V2(sin53degrees)
solve for V1 or V2,
V1 = V2(sin53degrees)/(sin37degrees)
plug this value into the first equation

X: 5 = [1.33 V2](cos53degrees) + V2(cos37degrees)

Solve for V2 to get 3.127
plug this into V1=1.33V2 to get the value for V1 and there you go!

To determine the final speed of each disk, we can use the conservation of momentum and the conservation of kinetic energy.

1) Conservation of Momentum:
The total momentum before the collision is equal to the total momentum after the collision. In this case, since the yellow disk is initially at rest, its momentum is zero.

Before the collision:
Momentum of orange disk = mass of orange disk * initial velocity of orange disk (p1 = m1 * vi)
Momentum of yellow disk = 0 (since it is initially at rest)

After the collision:
Let v1 be the final velocity of the orange disk and v2 be the final velocity of the yellow disk.

Momentum of orange disk = mass of orange disk * final velocity of orange disk (p1' = m1 * v1)
Momentum of yellow disk = mass of yellow disk * final velocity of yellow disk (p2' = m2 * v2)

Since the velocities are perpendicular after the collision, the direction of motion of the yellow disk changes completely, and its final velocity will be perpendicular to its initial velocity.

We can set up the momentum conservation equation as follows:
m1 * vi = m1 * v1 + m2 * v2

2) Conservation of Kinetic Energy:
To determine whether the collision is elastic or not, we need to check if the kinetic energy is conserved during the collision.

Before the collision:
Kinetic energy of orange disk = 0.5 * mass of orange disk * (initial velocity of orange disk)^2 (KE1 = 0.5 * m1 * vi^2)
Kinetic energy of yellow disk = 0 (since it is initially at rest)

After the collision:
Kinetic energy of orange disk = 0.5 * mass of orange disk * (final velocity of orange disk)^2 (KE1' = 0.5 * m1 * v1^2)
Kinetic energy of yellow disk = 0.5 * mass of yellow disk * (final velocity of yellow disk)^2 (KE2' = 0.5 * m2 * v2^2)

We can set up the kinetic energy conservation equation as follows:
0.5 * m1 * vi^2 = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2

Now we have two equations with two unknowns (v1 and v2). We can solve this system of equations to find the final velocities of each disk.

Simplifying the equations:
m1 * vi = m1 * v1 + m2 * v2
0.5 * m1 * vi^2 = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2

Dividing the second equation by m1:
0.5 * vi^2 = 0.5 * v1^2 + (m2/m1) * 0.5 * v2^2

Rearranging the first equation:
m1 * (vi - v1) = m2 * v2

Substituting m2 * v2 from the rearranged equation into the second equation:
0.5 * vi^2 = 0.5 * v1^2 + (m1 * (vi - v1) / m1) * 0.5 * v1^2

Simplifying further:
vi^2 = v1^2 + (vi - v1) * v1^2

Expanding the terms:
vi^2 = v1^2 + vi * v1^2 - v1^3

Rearranging the terms:
v1^3 - vi * v1^2 + vi^2 = 0

This is a cubic equation in terms of v1. To solve for v1, we can use numerical methods or algebraic methods depending on the given values.

Once we have v1, we can substitute it back into the momentum conservation equation to solve for v2.

To determine whether the collision is elastic or not, we need to check if the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

If the total kinetic energy is conserved (KE1 = KE1' + KE2'), then the collision is elastic.
If the total kinetic energy is not conserved, then the collision is inelastic.

To determine the final speed of each disk and whether the collision was elastic or not, we can use the principles of conservation of momentum and conservation of kinetic energy.

Let's denote the mass of each disk as m and the initial speed of the orange disk as vi.

Conservation of momentum:
In an isolated system, the total momentum before the collision is equal to the total momentum after the collision. Since the yellow disk is initially at rest, its momentum is zero. Therefore, the momentum of the system before the collision is equal to the momentum of the orange disk:

Before Collision: m * vi (momentum of orange disk) = m * v1 (momentum of orange disk after collision) + m * v2 (momentum of yellow disk after collision)

Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of a particle is given by 1/2 * m * v^2, where m is the mass and v is the speed of the particle.

Before Collision: (1/2) * m * vi^2 (kinetic energy of orange disk) = (1/2) * m * v1^2 (kinetic energy of orange disk after collision) + (1/2) * m * v2^2 (kinetic energy of yellow disk after collision)

Now let's solve these equations:

From the conservation of momentum equation:
m * vi = m * v1 + m * v2

Dividing both sides by m:
vi = v1 + v2 ---(Equation 1)

From the conservation of kinetic energy equation:
(1/2) * m * vi^2 = (1/2) * m * v1^2 + (1/2) * m * v2^2

Simplifying the equation and dividing both sides by (1/2) * m:
vi^2 = v1^2 + v2^2 ---(Equation 2)

Now let's use Equation 1 and Equation 2 to find the final speeds of the disks and whether the collision is elastic or not.

1) Final speeds of the disks:
We need to solve Equation 1 and Equation 2 simultaneously to find the values of v1 and v2.

From Equation 1: vi = v1 + v2

From Equation 2: vi^2 = v1^2 + v2^2

Rewriting Equation 2: vi^2 - v1^2 = v2^2

Substituting this into Equation 1: v1 = vi - v2

Now we substitute this value of v1 into Equation 2:

vi^2 - (vi - v2)^2 = v2^2

Simplifying this equation:
vi^2 - (vi^2 - 2 * vi * v2 + v2^2) = v2^2

vi^2 - vi^2 + 2 * vi * v2 - v2^2 = v2^2

2 * vi * v2 = 2 * v2^2

Cancelling out 2:
vi * v2 = v2^2

Dividing both sides by v2:
vi = v2

Therefore, the final speed of the yellow disk (v2) is equal to the initial speed of the orange disk (vi). The final speed of the orange disk (v1) can be found by substituting this value into Equation 1:

vi = v1 + v2
vi = v1 + vi
v1 = 0

Therefore, the final speed of the orange disk is 0, and the final speed of the yellow disk is equal to the initial speed of the orange disk.

2) Whether the collision was elastic or not:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. From the conservation of kinetic energy equation, we derived the equation vi^2 = v1^2 + v2^2.

Substituting the values we found above:
vi^2 = 0^2 + vi^2
vi^2 = 0 + vi^2
vi^2 = vi^2

This equation is true, meaning the total kinetic energy before and after the collision is the same. Therefore, the collision is elastic.

To summarize:
1) The final speed of the orange disk is 0, and the final speed of the yellow disk is equal to the initial speed of the orange disk (vi).
2) The collision is elastic.

thank you^