Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in a perfectly elastic glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving initially to the right at 5.60 m/s. After the collision, the orange disk moves in a direction that makes an angle of 33.4° with its initial direction. Meanwhile, the velocity vector of the yellow disk is perpendicular to the postcollision velocity vector of the orange disk. Determine the speed of each disk after the collision.

smaller speed______ m/s
larger speed_______ m/s

So far this is wut i tried,

Vo*cos33.4° + Vy*33.4° = 5.60
Vo*sin33.4° = Vy*33.4°

i don't if this make senses...i'm lost

You would ordinarily need to use both momentum and kinetic energy conservation to solve this problem, but they have told you the directions of both discs, so you have only two unknowns and can use two momentum-related equations. One of your two equations is not quite correct.

If Vo is the orange velocity after collision and Vy is the yellow velocity,
Vo sin 33.4 = Vy cos 33.4
and also
5.60 = Vo cos 33.4 + Vy sin 33.4

Plug in the trig function values and solve the two linear equations.

Vo = 1.516 Vy
5.60 = 0.8348 Vo + 0.5504 Vy
5.60 = 1.8259 Vy
Vy = 3.07 m/s
Vo = 4.65 m/s

Whenever there is elastic collision in a problem like this (one mass hits another stationary mass of equal mass), the paths of the two particles are at right angles, unless motion remains in the same direction. In that case, they exchange velocities. This can easily shown using the Pythagorean theorem.

To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.

Let's define the following variables:
- m: mass of each shuffleboard disk
- Vo: initial velocity of the orange disk (5.60 m/s)
- Vxo: x-component of the initial velocity of the orange disk
- Vyo: y-component of the initial velocity of the orange disk
- Vy: final velocity of the yellow disk
- Vx: final velocity of the orange disk
- θ: angle made by the final velocity vector of the orange disk with its initial direction (33.4°)

From the problem statement, we are given that the yellow disk is initially at rest and is struck by the orange disk moving initially to the right at 5.60 m/s. This means Vxo = Vo and Vyo = 0.

Using the principle of conservation of momentum, we have:
m * Vo = m * Vx + m * Vy

Using the principle of conservation of kinetic energy, we have:
(1/2) * m * Vo^2 = (1/2) * m * Vx^2 + (1/2) * m * Vy^2

Now, let's solve these equations to find the values of Vx and Vy.

From the equation of momentum conservation, we have:
Vo = Vx + Vy

Substituting the value of Vo, we get:
5.60 = Vx + Vy

From the equation of kinetic energy conservation, we have:
Vo^2 = Vx^2 + Vy^2

Substituting the value of Vo^2 and rearranging the equation, we get:
Vx^2 + Vy^2 = 5.60^2

Now, we have two equations:
1. 5.60 = Vx + Vy
2. Vx^2 + Vy^2 = 5.60^2

To solve these equations, we can use the trigonometric relationship between Vx, Vy, and θ. Since the velocity vector of the yellow disk is perpendicular to the post-collision velocity vector of the orange disk, we know that θ = 90°.

Using trigonometric relationships, we have:
Vx = V * cos(θ)
Vy = V * sin(θ), where V is the speed of each disk after the collision

Substituting these values in equations 1 and 2, we get:
5.60 = V * cos(θ) + V * sin(θ)
V^2 * (cos(θ)^2 + sin(θ)^2) = 5.60^2

Since cos(θ)^2 + sin(θ)^2 = 1, we can simplify the equation to:
V^2 = 5.60^2

Taking the square root of both sides, we get:
V = 5.60

Therefore, the speed of each disk after the collision is 5.60 m/s.

The smaller speed is 5.60 m/s, and the larger speed is also 5.60 m/s.