A particle of mass m falls from rest through a resistive medium having resistive force, F=-kv, where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity (v) versus time (t) ?

Solution

Physics sir · 4 steps

Drag grows as the particle speeds up, so the acceleration keeps dropping until v levels off.

1. Newton's second law. Take downward as positive. Gravity mg pulls down and the resistive force kv acts upward, against the motion.

   mdv/dt = mg - kv

2. Read the slope. At t = 0, v = 0, so the slope dv/dt = g is at its largest. As v grows, mg - kv shrinks, so the curve rises more and more gently. It becomes flat when kv = mg, at the terminal speed v_T = mg/k.
3. Exact solution. Integrating gives v = mg/k(1 - e^(-kt/m)). This starts at the origin and rises toward mg/k without ever crossing it.
4. So. The graph starts at (0, 0), is steepest at the start, and flattens to a constant value. That is graph (d).

The idea to keep. For drag F = -kv, write m dv/dt = mg - kv. The slope is largest at the start and falls to zero, so v rises and saturates at the terminal speed mg/k.

Answer from the NTA final answer key. Solution by PrepWiser.