Calculating trajectory of ball rolling down slope and off launch ramp
I am having some trouble with calculating the distance travelled and maximum height gained by balls rolling down a slope and off a lauch ramp.
There are 3 different types of ball, 100% steel, 100% nylon and 50% steel (by diameter) surrounded by 50% nylon. They are all 20mm diameter.
The balls are released at the top of a ramp of height h1 at angle a1 and they are launched from the end of a launch ramp of height h2 and angle a2. I have started tackling the problem by splitting it into two halves, one to calculate the velocity of the ball at the bottom of the first ramp and then to use this velocity as the start velocity of the second ramp trajectory calculation. For ease of calculation I have assumed that there is negligible loss of velocity through the curve that will obviously (in real life) sit between ramp 1 and ramp 2.
The balls will be rolling down and up the ramps so their accelerations (hence velocities) will be related to their moments of inertia, this is where I am having the trouble.
I think I will be fine working out the trajectory once I know the velocity at the end of the launch ramp (I am assuming air resistance is negligible).
Can anyone suggest where I might start looking for equations to solve the velocities at the end of each of the ramps?
Have I covered all the bases here or are there some other factors I have not taken into account?
Thanks in advance.