Collision of ball with inclined plane

[suryakalpo]

1. Hi,
I'm a computer engineering graduate. I'm developing a computer game similar to Soundrop (Apple). The situation is: balls fall vertically down under the effect of gravity. There are various inclined planes in the frame (of known angle of inclination). There could be multiple collisions possible and all collisions are considered to be perfectly elastic. I'm able to create a trajectory of flight for the balls till the next point of collision. However I'm unable to accurately determine the angle of reflection from the next incline.1

Homework Equations

I've used the following equations in my calculations of the trajectory:
x= x_initial + u_x_initial*T (T=time of flight till collision)
y=y_initial + u_y_initial*T - 0.5*g*T^2

I've also considered that the angle of incidence of a ball on the incline with respect to the normal to the plane is equal to the angle of reflection.
Thus for an incline with angle of inclination less than 90 (like this: ∠), I've considered the following:

θ= angle of inclination
γ= angle of incidence with respect to the horizontal
α= angle of reflection with respect to the horizontal

α= 2*θ+γ


Will this relation hold true for all values of θ and γ ? Or do I need to modify the equation for different cases ?

NOTE: I have used the normal x-y coordinate system and not oriented along the incline.

 

[BeBattey]

Is this in two dimensions or three?

Consider that conservation of linear momentum applies on the plane parallel to the surface the ball strikes.

Then if the collision is elastic the ball's speed is the same as right before the impact, and that linear momentum conservation can tell you one component of the velocity and the energy conservation can tell you the length, so you can extract the other component.

 

[suryakalpo]

Two dimensions

 

[suryakalpo]

I believe this holds when the coordinate system is along the inclined surface. Does it also hold true if the coordinate system is along the ground plane ?

 

[BeBattey]

Consider a ball striking normal to the surface. Then α = γ, and that's impossible with a non-zero θ, so I don't think that claim is true for all angles.

 

[BeBattey]

Linear momentum is conserved for any coordinate surface you're considering, but it's conserved along a line that's not with an axis making is frustrating to deal with is all. I don't believe you need to conserve momentum, as it's just the derivation of making the angle of incidence equal to the angle of reflection for elastic collisions.

I would set your angles with respect to the surface, then figure out what they are in terms of the angle of inclination from there.

For example for a surface of inclination θ, and a particle incident angle α(with the surface) the angle of reflection would be also α, so in terms of ground level α turns into (α+θ) and the reflection turns into (α-θ).

 

[suryakalpo]

So how do I calculate the angle of reflection with respect to the ground plane (not w.r.t to the incline). I'm reluctant to align the coordinate system along the incline because I do not understand how would the axes be if the angle of inclination of the plane is greater than 90 or negative with respect to the ground plane.

 

[suryakalpo]

I will illustrate my difficulty more specifically with an example. I've observed that for an incline (θ<90) and vector 'u' along the 4th quadrant, the ball can rebound off the plane or bounce off further in the same direction of motion depending on the orientation of the incline. Therefore in this case, I'm unsure about the sign to assign v_x. v_x as you know is the cos component and therefore the angle doesn't really determine the positive or negative direction of x-motion. How do I resolve this ?

 

[BeBattey]

Read the last part of my last response! I noted what they would be for ground level.

 

[suryakalpo]

I did read it. I just happened to have posted the message before I saw yours.
Thanks

 

[BeBattey]

Awesome, glad I could help :D

 

[suryakalpo]

I derived the angle of reflection for all four combinations of the directions of u_x and u_y and angle of inclination. However, I have a new problem now: If we consider the case of the first fall (free-fall under gravity) of any ball , it can correspond to the 'u' in the 3rd or 4th quadrant. But the angle of reflections in these two cases are different. One is γ+2θ and the other is γ-2θ, so in this case which angle should I assign to the velocity after collision ?