Classical Mechanics (Kibble & Berkshire) · Problem 4.17
Linear oscillations — T.W.B. Kibble, F.H. Berkshire
Physics — Mechanics · resisted-motion-quadratic-vertical-projection
Problem
A body is projected vertically upwards with speed and moves under uniform gravity in a medium that exerts a resistance force proportional to the square of its speed and in which the body's terminal speed is . Find the maximum height above the starting point attained by the body and the time taken to reach that height. Show also that the speed of the body when it returns to its starting point is . [Hint. The equations of motion for ascent and descent are different. See the note at the end of section 4.3.]Intuitive explanation
The key is recognizing that air resistance always opposes motion, so the equations of motion differ fundamentally on the way up versus down. On ascent, both gravity and drag pull downward, making deceleration fierce; on descent, drag opposes gravity, limiting speed to terminal velocity . Use energy or force balance to find where the body stops, then track the descent separately to see how much speed is lost to resistance by the time it returns.
Intuitive explanation
The key is recognizing that air resistance always opposes motion, so the equations of motion differ fundamentally on the way up versus down. On ascent, both gravity and drag pull downward, making deceleration fierce; on descent, drag opposes gravity, limiting speed to terminal velocity . Use energy or force balance to find where the body stops, then track the descent separately to see how much speed is lost to resistance by the time it returns.