First slide

Power

Question

A body is moving along a straight line under a force that delivers constant power. Distance travelled by the body in time ‘t’ is proportional to

Moderate

A

$t^{2}$
B

$t^{1 / 2}$
C

$t$
D

$t^{3 / 2}$

Solution

$P o w e r P = F \cdot v h e r e F i s f o r c e, v i s v e l o c i t y, g i v e n P o w e r i s c o n s \tan t ∴ constant = F . v ∴ F α \frac{1}{v} \Rightarrow m a α \frac{1}{v} m i s m a s s, a i s a c c e l e r a t i o n a α \frac{1}{v} \Rightarrow \frac{d v}{d t} α \frac{1}{v} \Rightarrow \frac{d v}{d t} \frac{d x}{d x} α \frac{1}{v} x i s d i s p l a c e m e n t, t i s t i m e \Rightarrow \frac{d x}{d t} \frac{d v}{d x} α \frac{1}{v} \Rightarrow \frac{v \cdot d v}{d x} α \frac{1}{v} \Rightarrow v^{2} . d v α d x i n t e g r a t i n g \Rightarrow \int v^{2} . d v α \int d x \Rightarrow \frac{v^{3}}{3} α x \Rightarrow v^{3} α x \Rightarrow v α x^{1 / 3} \Rightarrow \frac{d x}{d t} α x^{1 / 3} \Rightarrow \Rightarrow \frac{d x}{x^{\frac{1}{3}}} α dt i n t e g r a t i n g \Rightarrow \int \frac{d x}{x^{\frac{1}{3}}} α \int dt \Rightarrow \frac{x^{2 / 3}}{\frac{2}{3}} α t \Rightarrow x^{2 / 3} α t \Rightarrow x α t^{3 / 2}$

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