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I'm trying to plot a ball falling, hitting the ground and coming back up.

without air friction, the equations for the system would be

 eqn = {y''[t] == -9.8, y[0] == 10, y'[0] == 0}

Now I want to get the function so that each time $y(t)=0$, the velocity, $y'(t)$ will swap signs... This will mimic when the ball hits the ground

I tried :

 f = NDSolve[eqn, s, {t, 0, 20}, 
    Method -> {"EventLocator", "Event" -> s[t],
    "EventAction" :> s'[t]=-s'[t]}]

but alas, I get

"Set::write: Tag RuleDelayed in EventAction:>(s^[Prime])[t] is Protected. >>"


"NDSolve::bdmtd: The value of the option Method -> {EventLocator,Event->s[t],-(s^[Prime])[t]} is not a known built-in method, a symbol that could be a user-defined method, or a list with a name followed by method options. >>"

Any thoughts on what I could try?

Thanks in advance

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closed as too localized by Michael E2, m_goldberg, Simon Woods, Sjoerd C. de Vries, Yves Klett Apr 1 '13 at 17:53

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Have you seen the "Bouncing Ball" demo in the docs? –  J. M. Oct 18 '12 at 0:22
I had the same problem. It saddens me to say, but everything is right, except the last line: s'[t]->-s'[t] That is, a -> instead of =. –  user6669 Mar 31 '13 at 7:40
The first error is roughly a syntax error, missing parentheses: "EventAction" :> (s'[t]=-s'[t]). However, I don't think you can change y'[t] in the middle of the integration this way. You can stop integration and restart it as in the bouncing ball example J.M. refers to. –  Michael E2 Mar 31 '13 at 12:39
with v9, this works NDSolve[{y''[t] == -9.81, y[0] == 5, y'[0] == 0, WhenEvent[y[t] == 0, y'[t] -> -0.95 y'[t]]}, y, {t, 0, 10}] (again from the docs) –  acl Mar 31 '13 at 14:04

1 Answer 1

Why don't you just include a force to model the compression of the ball? Let's also provide parameters for a "spring constant" $k$, to which the force is directly proportional, and the ball radius $r$:

g[y_, k_, r_] := -9.8 + Piecewise[{{20, y <= 0}, {(1 - (y/r))^2 r k/y^2, 0 < y <= r}}, 0];

(The definition for $y\le 0$ is there for completeness. It shouldn't need to be used, but occasionally Mathematica's numerical searches take it to values of $y$ less than $0$, so supplying a net upward restoring force at points "below ground" gives it a useful hint concerning the physical intention.)

Manipulate lets you quickly explore the effect. It tracks the center of the ball.

 eqn = {y''[t] == g[y[t], Exp[k], r], y[0] == 10, y'[0] == 0};
 soln = NDSolve[eqn, y, {t, 0, 20}];
 Plot[(y /. First@soln)[t], {t, 0, 20}, AxesOrigin->{0, 0}, AxesLabel->{"Time(s)", "Height(m)"}],
 {{k, 8, "Log Spring"}, 0, 15}, {{r, 1, "Radius"}, 0.0001, 10}]

Elevation-time plot

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