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I'm trying to model a particle being dropped from a height onto a smooth inclined ramp, sliding down it and then freely falling after it leaves the ramp at the bottom (no friction or air resistance). The model should be as simple as possible.

There are 3 different stages of motion, which I guess need to be modelled separately.

I can use Newton's second law to calculate its acceleration while sliding (second stage of motion), and hence work out its velocity upon leaving the ramp (third stage of motion). Its motion is then that of a projectile, subject only to the force of gravity.

My question is about the first part of the motion, when the object is dropped onto the ramp. What would be the criteria for it to bounce? I'd like the model not to include bouncing. What should my assumptions be? What will be the initial conditions for the second stage of the motion, when the particle starts sliding down?

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If the particle does not bounce when it collides with the inclined plane then the component of its velocity perpendicular to the plane, when it hits the plane, is eliminated by the collision. Only the component of velocity parallel to the plane survives the collision.

If there is no friction or air resistance, then in each of the 3 stages - falling through the air, sliding down the ramp, falling through the air again - the mechanical energy (PE+KE) is conserved. ie $mgh+\frac12mv^2$ is constant. There is only a loss of energy when the particle collides with the ramp.

In the final stage the particle behaves as a projectile and follows a parabolic path.

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  • $\begingroup$ Or you could have an "on-ramp" - a curve that starts vertical, and curves around so the initial vertical velocity of the object gets converted to velocity along the ramp... $\endgroup$ – Floris Feb 14 '17 at 3:27
  • $\begingroup$ Thank you! In this instance, the ramp just need to be a classic straight slope, but it's a clever idea. Now assume there is actually friction between the particle and the ramp, but we don't know the material the particle is made of (we do know what the ramp is made of). Is there any way to work out some sort of "average" coefficient of sliding/kinetic friction? $\endgroup$ – Nerd Feb 16 '17 at 9:54
  • $\begingroup$ You can measure the coefficient of sliding friction by tilting the ramp until the particle slides down it at constant speed. You should give it a slight push to start it going, because static friction is usually greater than sliding friction. The coefficient is then $\tan\theta$ where $\theta$ is the angle between the slope and the horizontal. It is actually easier to calculate the ratio of vertical and horizontal distances than to measure the angle. $\endgroup$ – sammy gerbil Feb 16 '17 at 17:31

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