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

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Laws of Motion

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)?

Answer: (B)

The equation of motion of the particle falling from rest is

m \frac{d v}{d t} = m g - k v .

Separating variables and integrating from rest,

\int_{0}^{v} \frac{d v}{m g - k v} = \int_{0}^{t} \frac{d t}{m},

giving

v = \frac{m g}{k} (1 - e^{- k t / m}) .

t = 0

the speed is zero, and as

t \to \infty

it approaches the terminal value

v_{t} = m g / k

. The velocity therefore starts from the origin, rises with a continuously decreasing slope, and saturates towards the horizontal line

v = m g / k

, which is the exponential growth curve that begins at the origin and flattens to a constant terminal velocity.

JEE Main · 2026 · 28 Jan 2026 · Shift 1 · Single correct option