Question 184: Properties of Matter

Question

Which of the following option correctly describes the variation of the speed v and acceleration ‘a’ of a point mass falling vertically in a viscous medium that applies a force F = − kv, where ‘k’ is a constant, on the body ? (Graphs are schematic and not drawn to scale)

Accepted Answer

Equation of motion for the mass,
ma = mg $$-$$ kv
$$ \Rightarrow $$    $${{dv} \over {dt}} = {{mg - kv} \over m}$$
$$ \Rightarrow $$   $$\int\limits_0^v {{{dv} \over {mg - kv}}}  = {1 \over m}\int\limits_0^t {dt} $$
$$ \Rightarrow $$   $$ - {1 \over k}\left[ {\ln \left( {mg - kv} ight)} ight]_0^v = {t \over m}$$
$$ \Rightarrow $$   $$\ln \left( {{{mg - kv} \over {mg}}} ight) =  - {{kt} \over m}$$
$$ \Rightarrow $$   $$1 - {{kv} \over {mg}} = {e^{ - {{kt} \over m}}}$$
$$ \Rightarrow $$   $${{kv} \over {mg}} = 1 - {e^{ - {{kt} \over m}}}$$
$$ \Rightarrow $$   $$v = {{mg} \over k}\left( {1 - {e^{ - {{kt} \over m}}}} ight)$$
ma $$=$$ mg  $$-$$ k $$ 	imes $$ $${{mg} \over k}$$ (1 $$-$$ e$$^{ - {{kt} \over m}}$$)
$$=$$  mg $$-$$ mg  +  mge$$^{ - {{kt} \over m}}$$
a  $$=$$  g e$$^{ - {{kt} \over m}}$$

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Properties of Matter

Which of the following option correctly describes the variation of the speed v and acceleration ‘a’ of a point mass falling vertically in a viscous medium that applies a force F = − kv, where ‘k’ is a constant, on the body ? (Graphs are schematic and not drawn to scale)

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