A tennis ball is tossed vertically upward with initial velocity m/s from ground level
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A tennis ball is tossed vertically upward with initial velocity v0 m/s from ground level. Air resistance exerts a force that experimental data suggest is proportional to the instantaneous velocity of the ball. Neglect all forces except gravity and air resistance.
Let v(t) denote the velocity (in m/s) of the ball at time t seconds after it leaves the player’s hand. Take the upward direction as positive.
(a) Derive a differential equation satisfied by v(t) that includes a proportionality parameter. Do not attempt to solve for v(t).
(b) Discuss the required sign of the parameter so that the modeled air resistance opposes the direction of motion throughout the flight. Use separate arguments for the upward and downward portions of the trajectory.
(c) Explain qualitatively how increasing the magnitude of the parameter affects the maximum height reached by the ball, assuming all other conditions are unchanged.