A particle of mass mmm 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 t

Question

A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t)? [2026]

Smart Achievers · Accepted Answer

(2)

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

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

$- \frac{1}{k} l n (\frac{m g - k v}{m g}) = \frac{t}{m}$

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

Solution

Q.
A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t) [2026]

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Ans.
(2)

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

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

$- \frac{1}{k} l n (\frac{m g - k v}{m g}) = \frac{t}{m}$

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

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Solution

Q. 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) [2026]

1

2

3

4

Ans. (2) m·dvdt=mg-kv ∫0vdvmg-kv=∫0tdtm -1kln(mg-kvmg)=tm v=mgk(1-e-kt/m)

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Q.
A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = - k v$ , where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity (v) versus time (t) [2026]

Ans.
(2)

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

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

$- \frac{1}{k} l n (\frac{m g - k v}{m g}) = \frac{t}{m}$

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