Question 3.

a)  A ball with mass m is projected vertically upward from the earth's surface with a positive initial velocity v0 We assume the forces acting on the ball are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude p|vt|, where p is a positive constant and v(t) is the velocity of the ball at time t. In both the ascent and the descent, the total force acting on the ball is -pv-mg. So, by Newton's Second Law, the equation of motion is: mv' = -pv-mg Solve this differential equation to find v(t).

b)  Find the height of the ball, y(t), as a function of time until it hits the ground (recall velocity is the derivative of displacement)

c)  Let t1 be the time that ball takes to reach its maximum height Show that

Find t1 for a ball with mass 2 kg and initial velocity of 30 m/s. Assume air resistance is 1/2 of the speed.

d)  Determine whether y(2t) is positive or negative. Use this to show whether ascent or decent is faster?