# Math Help - Collisions (physics problem)

1. ## Collisions (physics problem)

In a game of marbles, a purple marble makes an off-center collision with a red marble of the same mass. Before the collision, the purple marble travels in the positive x direction and the red marble is at rest. After the collision, the purple marble's velocity of 0.0900 m/s makes a 35.0 degree angle above the positive x axis and the red marble's velocity makes an angle of negative 20.0 degrees with the axis. Find the initial speed of the purple marble.

I think this is a completely elastic collision which means the kinetic energy is conserved. The momentum is also conserved. I'm a bit lost on how to do the problem because it doesn't tell me what is the red marble's final velocity (maybe I don't need it...).

Thank you!

2. I'm not entirely sure if this is correct, but if it's not, it's at least a step in the right direction.

First, you shouldn't need the red marble's final velocity because you need conserve energy in both the x and y dimensions, which will allow you to set up a system of equations. This system will have 2 unknowns, so you can just express the red marble's velocity in terms of the purple's.

First, you want to set it up so that the energy equation in the x direction is as follows (P.S. I factored out, and divided the 1/2*m factor because it is a like factor)

$\vec{V_{1}}^2=(\vec{V_{2}}\cos(35))^2+(\vec{V_{3}} \cos(20))^2$

Doing the same for the y dimension, we get the following

$0=(\vec{V_{2}}\sin(35))^2-(\vec{V_{3}}\sin(20))^2$

Using the latter, we can express $\vec{V_{3}}$ in terms of $\vec{V_{2}}$ by solving for $\vec{V_{3}}$

You should get $\vec{V_{3}}=\vec{V_{2}}\frac{\sin(35)}{\sin(20)}$

When we replace the $\vec{V_{3}}$ with the solution we just got, we get the equation $\vec{V_{1}}^2=(\vec{V_{2}}\cos(35))^2+(\vec{V_2}\c ot(20)\sin(35))^2$

or

$\vec{V_{1}}=\sqrt{(\vec{V_{2}}\cos(35))^2+(\vec{V_ 2}\cot(20)\sin(35))^2}$

3. Originally Posted by Linnus
In a game of marbles, a purple marble makes an off-center collision with a red marble of the same mass. Before the collision, the purple marble travels in the positive x direction and the red marble is at rest. After the collision, the purple marble's velocity of 0.0900 m/s makes a 35.0 degree angle above the positive x axis and the red marble's velocity makes an angle of negative 20.0 degrees with the axis. Find the initial speed of the purple marble.

I think this is a completely elastic collision which means the kinetic energy is conserved. The momentum is also conserved. I'm a bit lost on how to do the problem because it doesn't tell me what is the red marble's final velocity (maybe I don't need it...).

Thank you!
I don't have much time for right now, but I can send you down the right path.

Momentum is conserved in both the x and y directions (using the typically coordinate system.) So we have that, in the x direction:
$mv_{p0} = mv_{px} + mv_{rx}$

or
$mv_{p0} = m(0.0900)~cos(35) + mv_r~cos(20)$

and in the y direction:
$0 = mv_{py} + mv_{ry}$

or
$0 = m(0.0900)~sin(35) - mv_r~sin(20)$

As this is enough to solve the system, and we know that momentum has to be conserved, we should not assume that energy is.

-Dan

4. Ah yes...I totally meant to do my post assuming conserving momentum, not energy.