Ball Bouncing on Hilly Terrain
The following application demonstrates how event modeling in the dsolve command can be used to model a ball bouncing on a hilly terrain.
restart;
withplots:
Define the Surface
We define the surface, surf, in the line below. It is possible to modify this equation to show different "terrains".
surf≔ sinx+0.2cos4x+ sin4x−0.2 x+3:
plotsurf,x=0..10,scaling=constrained
Derive the Hilly Surface
The velocity vector of a ball:
V≔Vectorx.t,y.t:
The surface normal:
N≔LinearAlgebra:−NormalizeVector−∂∂xsurf,1,2,conjugate=falseassuming positive:
N≔evalN,x=xt
The velocity vector after reflection across the surface normal:
V_reflect≔−1+C__R V.NN+V:
V_reflect:=simplifyV_reflect,symbolic:
Restitution coefficient:
C__R ≔ 0.99:
Using C__R < 1 represents an inelastic collision between the ball and the surface, whereas using C__R =1 represents an elastic collision.
Differential Equations and Initial Conditions
Gravity acts in the -y direction:
deqs≔y..t=−9.81,x..t=0:
ics≔Dx0=0,Dy0=0,x0=2,y0=4.5:
Solve and Animate the Differential Equations
sol≔dsolvedeqs,ics,xt,yt,numeric, events=yt=evalsurf,x=xt,temp=x.t,x.t= V_reflect1,y.t= subsx.t=temp,V_reflect2, range=0..10, output=listprocedure:
Animate the Ball Bouncing on the Terrain
xanim≔subssol,xt:yanim≔subssol,yt:
p1≔plotsurf,x=0..10,color=black,filled=true, transparency=0.5:p2≔animatepointplot,xanimt,yanimt,symbol=solidcircle,symbolsize=15,color=black,t=0..10,frames=150:
displayp1,p2,view=0..10,0..5,scaling=constrained
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