Ballistics and Quadratics

A problem from Stewart Calculus: Concepts and Contexts 2/e p. 30.


Ball = fetchData("PREP-Toronto-ball.csv")
modHeight = fitModel(Height ~ a * Time^2 + b * Time + c, data = Ball)
findZeros(modHeight(t) ~ t, t.lim = c(5, 15))
## [1] 9.674

An a priori model:

apriori = makeFun(450 - 0.5 * 9.806 * Time^2 ~ Time)

How good are the models?

aprioriResids = with(data = Ball, Height - apriori(Time))
fittedResids = with(data = Ball, Height - modHeight(Time))
plotPoints(fittedResids ~ Time, data = Ball)

plot of chunk unnamed-chunk-3


Do some research on “terminal velocity” to figure out what would be an appropriate terminal velocity for a ball. One equation you may find is \[ V_t = \sqrt{\frac{2 m g}{\rho A C_d}}, \] where \( m \) is the mass of the ball, \( g \) the acceleration due to gravity, \( \rho \) the density of air, \( A \) the cross-sectional area of the ball, and \( C_d \) the “drag coefficient.” You should be able to find values for each of these and a plausible range of values for \( m \) and \( A \) for a “ball.” For a sphere, the drag coefficient is \( C_d = 0.47 \).

Going further

Build a differential equation with a reasonable model of air drag. In particular, arrange it to reach a plausible terminal velocity. How far is height versus time from a parabola?

Here's a video of a falling ball. Perhaps you can find other data for longer falls. Try to find some compelling data and see if they display terminal velocity.