# Ball hitting an inclined plane

## Homework Statement

http://www.sumoware.com/images/temp/xzafcttptkeoehmo.png [Broken]
An inclined plane is put on a smooth floor. The inclined plane is hit (collided) by an elastic ball moving horizontally before the collision. The ball bounce from the inclined plane and land again right at the point of first collision. If the inclination angle is Θ, determine the ratio between the ball mass and the inclined plane mass

px = px'
E = E'

## The Attempt at a Solution

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Actually I don't know what to do, so I just draw the condition after collision
http://www.sumoware.com/images/temp/xzknipxthmfhalxb.png [Broken]
Vp is the velocity of inclined plane

p = p
mb vb = mb vbx' + mp vp'

Then, I don't know what to do

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ehild
Homework Helper
Energy is conserved, as the ball is elastic and the ground is smooth.
The ball reaches the plane at the first collision point again. What does it mean for the horizontal distances travelled by the plane and by the ball? What information you have about the vertical and horizontal velocity components of the ball?

What does it mean for the horizontal distances travelled by the plane and by the ball?
Hmm... When the ball reaches the plane at the first collision point, it means that the horizontal distance travelled by the ball is the same as travelled by the plane.
So
p = p
mb vb = mb vbx' + mp vp'
mb vb = (mb + mp)v'x

What information you have about the vertical and horizontal velocity components of the ball?
The vertical component of the ball is v'by = v cos Θ
Horizontal component is v'bx = v sin Θ

Energy is conserved, as the ball is elastic and the ground is smooth.
(1/2)mvb2 = (1/2)mvb'x2 + (1/2)mvp'x2
(1/2)mvb2 = mvb'x2
vb2 = 2vb'x2
vb2 = 2(vb sin Θ)2
vb = √(2(vb sin Θ)2)

ehild
Homework Helper
Hmm... When the ball reaches the plane at the first collision point, it means that the horizontal distance travelled by the ball is the same as travelled by the plane.
So
p = p
mb vb = mb vbx' + mp vp'
mb vb = (mb + mp)v'x
Correct.
The vertical component of the ball is v'by = v cos Θ
Horizontal component is v'bx = v sin Θ
Why??????

Correct.

Why??????
Because if theta increases, the y component of the velocity of ball after collision decreases, right ? So, it's cosine function, right ?
And if theta increases, the x component of the velocity of ball after collision increases, right ? So, it's sinus function, right ?
But, I don't know what it will be if the 'launch' angle has nothing to do with the angle theta

haruspex
Homework Helper
Gold Member
Because if theta increases, the y component of the velocity of ball after collision decreases, right ? So, it's cosine function, right ?
And if theta increases, the x component of the velocity of ball after collision increases, right ? So, it's sinus function, right ?
But, I don't know what it will be if the 'launch' angle has nothing to do with the angle theta
There are trig functions besides cos and sine.
Let the impulse the block delivers to the ball be J. Which way does it point? What is its component in the x direction?

http://www.sumoware.com/images/temp/xzaciotcpqkxexdr.png [Broken]
J = mg cos Θ Δt
Jx = m v'bx - m vb

But, what is v'bx ?

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ehild
Homework Helper
The plane can exert force only in the perpendicular direction, as you drew, but the impulse is not connected to gravity. The parallel (with the plane) component of the initial velocity of the ball does not change during the collision. Only the perpendicular component changes. Conservation of energy provides a relation among the speed of the slope Vp, initial speed of the ball vb, and the perpendicular component after collision, vm.

Both the perpendicular and parallel components of the velocity after collision have horizontal components, depending on the angle theta. You can get an expression for v'bx in terms of theta.
You know that v'bx=Vp, and conservation of momentum yields Vp=mbvb/(mb+mp). There are enough equation to get the mass ratio in terms of theta.

haruspex