Chapter 3.9, Problem 37E

Problems 35–37 investigate the motion of a projectile shot from a cannon. The fixed parameters are the acceleration of gravity, g = 9.8 m∕sec², and the muzzle velocity, υ₀ = 500 m∕sec, at which the projectile leaves the cannon. The angle θ, in degrees, between the muzzle of the cannon and the ground can vary.

At its highest point the projectile reaches a peak altitude given by

$h(\theta )=\frac{{\upsilon }_0^2}{2g}{\sin }^2\frac{\pi \theta }{180}=12,755\sin \frac{\pi \theta }{180}\text{meters}.$

1. (a) Find the peak altitude for θ = 20°.
2. (b) Find a linear function of θ that approximates the peak altitude for angles near 20°.
3. (c) Find the peak altitude and its approximation from part (b) for 21°.