Ellastic collision between a frame and projectile

[Rosengrip]

Homework Statement

To rods, each with mass 0.3 kg and length 0.2 m, are welded together at one of their ends, so they form an L-shaped frame.

A ball with mass 0.3 kg and with speed 5 m/s hits one of the rods at its end at an angle of 0 degrees (so ball speed vector is perpendicular to one of the rods and parallel to the other) and rebounds (elastic collision).
Image represents the event:

[PLAIN]http://www.shrani.si/f/0/Dw/12MBWrLL/okvir.jpg

What's the speed of center of mass of frame and ball after the collision?
What's the angular speed of frame after collision?

Homework Equations

I=(1/12)ml²
Moment of inertia equations, mechanical energy equations, linear momentum equations

The Attempt at a Solution

Firstly I calculated the position of center of mass (COM from now on)and moment of inertia of frame if it's spinning around it.
If I put the center of coordinate system in the welding point, the COM is at coordinates (-0.05, -0.05) (point T on the picture).

Using parallel axis theorem I get the moment to be I=(5/12)ml² (m=0.3 kg, l=0.2m). I=0.005 kgm²

Because the collision is elastic, linear momentum, mechanical energy and angular momentum of ball and frame combined stay the same.

I figured because the ball doesn't strike at an angle perpendicular to the vector pointing from COM to the strike point, we get 2 sets of equations.
1 component of ball speed vector (Vx) moves and rotates the frame (the one perpendicular to vector from COM to impact point).
1 component of ball speed vector (Vy) only moves the frame (parallel to the vector).
[PLAIN]http://www.shrani.si/f/I/FW/migS8fu/okvir.jpg

Let the component of speed vector perpendicular to vector from COM to impact point be Vx, the one which is parallel Vy.

m= ball mass = 0.3 kg
M= frame mass = 0.6 kg
Vx = ball speed x component before collision
Vy = ball speed y component before collision
Vx1 = ball speed x component after collision
Ux1 = frame speed x component after collision
Vy1 = ball speed y component after collision
Uy1 = frame speed y component after collision
I =f rame moment of inertia
w = angular speed of frame
d = distance between COM and impact point

Set of equations for Vx:
m(Vx) = -m(Vx1) + M(Ux1) -linear momentum
m(Vx) = -m(Vx1)d + Iw - angular momentum
(1/2)m(Vx)²=(1/2)m(Vx1)²+(1/2)M(Ux1)²+(1/2)Iw² - mechanical energy (kinetic and rotational energy)

Set of equations for Vy:
m(Vy) = -m(Vy1) + M(Uy1) -linear momentum
(1/2)m(Vy)²=(1/2)m(Vy1)²+(1/2)M(Uy1)² - mechanical energy (kinetic and rotational energy)

5 equations, 5 unknown variables (Vx1, Vy1, Ux1, Uy2, w). Threw the whole mess into Mathematica and I got some results.
The thing is, I'm not sure if this is the correct way since I'm new to this field. Any info on this would be greatly appreciated.
If there's any info missing, please tell me :)

 

[gneill]

The COM of the overall system won't be the same as the COM of the two-rod assembly, since the latter doesn't account for the mass and position of the ball.

 

[Rosengrip]

Uh now that further complicates things doesn't it. I have a similar case with only 1 rod (COM at the center of rod) which gets hit by a ball and there ball doesn't have anything to do with COM.
I can't see how ball can affect the change of COM :/. It's not connected in any way to the frame, it only gives its momentum to the frame which then starts traveling with some velocity and spinning around its COM at the same time.

 

[gneill]

Uh now that further complicates things doesn't it. I have a similar case with only 1 rod (COM at the center of rod) which gets hit by a ball and there ball doesn't have anything to do with COM.
I can't see how ball can affect the change of COM :/. It's not connected in any way to the frame, it only gives its momentum to the frame which then starts traveling with some velocity and spinning around its COM at the same time.

It's the center of mass of the whole system that moves inertially. Otherwise you're dealing with external forces, and all your conservation laws go out the window.

 

[Rosengrip]

Yeah I guess that's correct. So I'm pretty much lost right now. Any tips on how to get the new COM? :)

 

[gneill]

Yeah I guess that's correct. So I'm pretty much lost right now. Any tips on how to get the new COM? :)

I came across an interesting web page a while back that showed how to do collisions involving rotations. It took me a while to find it again

Have a look here:

http://www.myphysicslab.com/collision.html"

 

[Rosengrip]

Thanks for the site, it's pretty interesting.
I guess the most confusing thing here is that I have a solved case (ball hitting a rod) in which the COM stays the same (at the center of the rod) throughout the whole process (before and after the collision). The conservation laws still apply there. My case is pretty similar, the only different thing is the angle between initial ball velocity vector and COM-impact point vector.

 

[Rosengrip]

Since I can't edit my first post here, I just want to correct my question:
I'm looking for a speed of COM of frame after collision AND speed of ball after collision, not the speed of COM of frame and ball. I hope that's clearer:>