382: Math Scientic Computing Trajectory of a Spinning Tennis Ball Goal: This hand out formulates an ODE model to describe the trajectory of a spinning tennis ball and discusses how to calculate its solution under different assumptions (e.g., vacuum vs. air resistance) with MATLAB. The Model m mass of the ball angular velocity, a vector in the direction of the axis of rotation.
1
consult gure presented in class
regard ball as a mass point (why is this ok? after all, we are considering the effects of the air in the motion of the ball, size and surface should matter... we are not really ignoring those...) moving under the inuence of the forces: weight drag G = mg, g = (0, 0, g) D = DL (v) v , v opposite to velocity orthogonal to and v
Magnus force
M = ML v , v
The magnitudes of the drag and Magnus forces are given by 1 d2 2 v 2 4 1 d2 2 v 2 4
DL (v) = CD ML (v) = CM density of air
(1) (2)
CD , CM - determined experimentally, they depend on velocity, ball revolution and material of surface (it is ok to assume point mass) Experimental data for tennis ball suggests:
Math 382: Scientic Computing
2
CD = 0.508 + 1 22.053 + 4.196
v w
5 2
2 5
(3)
CM = where w =
d 2
1 2.022 + 0.981
v w
(4)
| v | v
Newtons second law trajectory denes of the ball
v v d2 r(t) = mg DL (v) + ML (v) m 2 dt v v with initial conditions r(0) = r0 and dr (0) = v0 dt
(5)
Most usual case is the topspin lob lies on horizontal plane and is orthogonal to v0 (so by equation (5) also orthogonal to v(t)). This means motion is on vertical plane, so we chose x-axis to run along that plane, (5) becomes x = CD v x + CM v z z = g CD v z CM v x
d2 , 8m
(6) (7)
where v =
x2 + z 2 and =
is direction of rotation, = 1 for topspin
Initial conditions: x(0) = 0, z(0) = h, x(0) = v0 cos , z(0) = v0 sin (8)
where v0 is the magnitude of the initial velocity vector v0 and the angle between v0 and the x-axis. Case I: motion in vacuum x = 0, with the above initial condi...