Chapter 11.8, Problem 63E

Projectile trajectories A projectile (such as a baseball or a cannonball) launched from the origin with an initial horizontal velocity u₀ and an initial vertical velocity v₀ moves in a parabolic trajectory given by

$x=u_{0}t,y=-\frac{1}{2}g{t}^2+v_{0}t,\text{for}t\ge \text{0}$

where air resistance is neglected and g ≈ 9.8 m/s² is the acceleration due to gravity (see Section 11.7).

1. a. Let u₀ = 20 m/s and v₀ = 25 m/s. Assuming the projectile is launched over horizontal ground, at what time does it return to Earth?
2. b. Find the integral that gives the length of the trajectory from launch to landing.
3. c. Evaluate the integral in part (b) by first making the change of variables u = −gt + v₀. The resulting integral is evaluated either by making a second change of variables or by using a calculator. What is the length of the trajectory?
4. d. How far does the projectile land from its launch site?