First slide

Power

Question

The speed v reached by a car of mass m in travelling a distance x, driven with constant power P, is given by

Moderate

A

$v = \frac{3 x P}{m}$
B

$v = {(\frac{3 x P}{m})}^{\frac{1}{2}}$
C

$v = {(\frac{3 x P}{m})}^{\frac{1}{3}}$
D

$v = {(\frac{3 x P}{m})}^{2}$

Solution

powere =P= Fv here F is force, v is velocity, and F= ma, here a is acceleration
P =m $\frac{d v}{d t}$ v, here differential form of acceleration is considered
or v $\frac{d v}{d t}$ = $\frac{P}{m}$
or v $\frac{d v}{d x} \frac{d x}{d t} = \frac{P}{m}$ , here in left hand side , divide and multiply by dx,
or $v^{2} \frac{d v}{d x} = \frac{P}{m}$ or $v^{2} d v = \frac{P}{m} d x$

$\int v^{2} d v = \int \frac{P}{m} d x$
On integration , we get $\frac{v^{3}}{3} = \frac{P x}{m}$ or v = ${(\frac{3 x P}{m})}^{\frac{1}{3}}$

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