1. a. Suppose that a ball is thrown vertically with gravity as the only force acting on this ball (air resistence is ignored). From Newton's law of motion, the mass times acceleration of the ball is equal to this force of gravity. Thus, ma = -mg, where m is the mass of the ball, a(t) is the acceleration of the ball, and -mg is the force of gravity (assuming up is positive), where g = 9.78 m/sec2. Since acceleration is the derivative of the velocity and velocity is the derivative of position, then if h(t) is the height of the ball at any time t, then a(t) = h''(t). It follows that the height of a ball is governed by the second order differential equation

h''(t) = -g.

Suppose that the ball is thrown vertically with a velocity of 12.8 m/sec from a position on the ground. This means that the initial velocity is given by v(0) = h'(0) = 12.8 m/sec and the initial position is h(0) = 0.

Solve this differential equation to find the velocity, v(t), and height, h(t), of the ball for any time up until the ball hits the ground. Determine the velocity and the height of the ball at t = 1 and 2 sec. Find the maximum height of the ball and when this occurs. Also, determine the time and velocity of the ball when it hits the ground. Draw a graph of the height of the ball as a function of time for all times up until the ball hits the ground.

b. The previous problem uses simple integration to find the height of the ball. Below is a collection of indefinite integrals that you can use Maple to solve. Write the solution of the integral and a brief description of the rules that give you the answer based on your observations.