Example 2: The Length of the Path of a Ball

Figure 6.3.5

Find the length of the path of a ball whose motion is given by

x = 20t,       y = 32t - 16t²

from t = 0 until the ball hits the ground. (Ground level is y = 0, see Figure 6.3.5.) The ball is at ground level when

32t - 16t² = 0, t = 0 and t = 2.

We have

dx/dt = 20, dy/dt = 32 - 32r,

We cannot evaluate this integral yet, so the answer is left in the above form. We can get an approximate answer by the Trapezoidal Rule. When Δx = 1/5, the Trapezoidal Approximation is

s ~ 53.5 error < 0.4.