Bouncing Ball II

Here, the ball hits a (45 degree) slope.

If the (point-like) ball is at some position (x,y) below the slope, as shown below, then I am assuming an elastic restoring force acting along the normal to the slope, and that this force is proportional to the distance q, according to Hooke's law:

Hence, if we are below the slope (i.e. x > y), we have (where m is mass of ball, k the spring constant of the slope and g the acceleration due to gravity):

And if the ball is above the slope, then we just have gravity acting (along the negative y-axis) on the ball. This is easy enough to patch up:

The right hand side summing amplifier (10x gain) has both a negative (i.e. inverting) and positive (i.e. non-inverting output, both used here. The switch closes every time the ball goes below the surface - thus applying the restoring acceleration. The left integrators also have gains of 10x, and hence k/(2m) = 100 and if we assume a mass of 1 kg, we thus have a spring constant of 200 N/m.

2 V/cm both axes - ball starts at (4,6)

2 V/cm both axes - ball starts at (5,6). Second bounce visible.

The upper trace shows the ball dropping vertically until it hits the slope and then bouncing of the slope. The lower trace shows what happens when the ball is dropped closer to the slope (i.e. further to the right) - the ball has less kinetic energy when it hits the slope the first time and thus makes a lesser bounce. The ball comes down to hit the slope a second time (at about x = -3). In all cases the ball sinks into the slope somewhat - due to the finite and slightly squidgy (200 N/m) spring constant which was assumed.