First slide

Power

Question

A machine delivers power to a body which is proportional to velocity of the body. If the body starts with a velocity which is almost negligible, then the distance covered by the body is proportional to

Moderate

A

$\sqrt{v}$
B

$\sqrt[3]{\frac{v}{2}}$
C

$v^{5 / 3}$
D

v²

Solution

$\begin{array}{l} Power = Fv \propto v \\ \Rightarrow mv \frac{dv}{dt} = kv, where k = constant \\ \Rightarrow m \frac{dv}{dt} = k \Rightarrow mv \frac{dv}{dx} = k \\ \Rightarrow vdv = \frac{k}{m} dx \end{array}$
Integrating both sides, we get $\frac{v^{2}}{2} = \frac{k}{m} x$
Hence, displacement is proportional to the square of instantaneous velocity.

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