Question 64: Laws of Motion

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 $(t)$ ?

Accepted Answer

As the particle falls, two forces act on it :
Weight $\left(\mathrm{F}_{\mathrm{g}}=\mathrm{mg}ight)$ acting downwards.
Resistive Force ( $\mathrm{F}=\mathrm{kv}$ ) acting upwards (opposing motion).
The net force on the particle is,
$$ \mathrm{F}_{\mathrm{net}}=\mathrm{mg}-\mathrm{kv} $$
According to Newton's Second Law, $\mathrm{F}_{\mathrm{net}}=\mathrm{m} \cdot \mathrm{a}=\mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}}$. Therefore
$$ \mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{mg}-\mathrm{kv} $$
$$\Rightarrow $$ $$ \frac{\mathrm{dv}}{\mathrm{mg}-\mathrm{kv}}=\frac{1}{\mathrm{~m}} \mathrm{dt} $$
Integrate both sides, knowing that at $\mathrm{t}=0, \mathrm{v}=0$ (starting from rest) :
$\int\limits_0^{\mathrm{v}} \frac{\mathrm{dv}}{\mathrm{mg}-\mathrm{kv}}=\frac{1}{\mathrm{~m}} \int\limits_0^{\mathrm{t}} \mathrm{dt}$
$$\Rightarrow $$ $-\frac{1}{\mathrm{k}}[\ln (\mathrm{mg}-\mathrm{kv})]_0^{\mathrm{v}}=\frac{\mathrm{t}}{\mathrm{m}}$
$$\Rightarrow $$ $\ln \left(\frac{\mathrm{mg}-\mathrm{kv}}{\mathrm{mg}}ight)=-\frac{\mathrm{k}}{\mathrm{m}} \mathrm{t}$
$$\Rightarrow $$ $\frac{m g-k v}{m g}=e^{-\frac{k}{m} t}$
$$\Rightarrow $$$1-\frac{\mathrm{kv}}{\mathrm{mg}}=\mathrm{e}^{-\frac{\mathrm{k}}{\mathrm{m}} \mathrm{t}}$
$$\Rightarrow $$ $$ v(t)=\frac{m g}{k}\left(1-e^{-\frac{k}{m} t}ight) $$
The resulting equation $\mathrm{v}(\mathrm{t})=\mathrm{v}_{\mathrm{t}}\left(1-\mathrm{e}^{-\frac{\mathrm{t}}{	au}}ight)$ where $\mathrm{v}_{\mathrm{t}}=\frac{\mathrm{mg}}{\mathrm{k}}$ is the terminal velocity.
At $\mathrm{t}=0, \mathrm{v}=\mathrm{v}_{\mathrm{t}}(1-1)=0$. Hence, the graph must start at the origin $(0,0)$.
As $\mathrm{t} ightarrow \infty$ the term $\mathrm{e}^{-\frac{\mathrm{k}}{\mathrm{m}} \mathrm{t}}$ approaches 0 , so v approaches a constant value $\frac{\mathrm{mg}}{\mathrm{k}}$ (terminal velocity).
This is an exponential graph with a horizontal asymptote at $\mathrm{v}=\frac{\mathrm{mg}}{\mathrm{k}}$
Hence, the correct option is (D).

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Laws of Motion

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) ?

Choose the correct answer: