Question

A body is moved from rest along a straight line by a machine delivering constant power. The distance moved by the body in time $t$ is proportional to

• A. $t^{1/2}$
• B. $t^{3/4}$
• C. $t^{3/2}$
• D. $t^{2}$

Answer 1

Answer: (C)

$t^{3/2}$

Answer 2

Power $P=\frac{d W}{d t}=\frac{F \cdot d s}{d t}$ ( $\therefore d W=F \cdot d s$ ) $=m \frac{d v}{d t} \cdot \frac{d s}{d t}=\frac{m v d v}{d t}$ $\therefore P$ is constant, say $K$ then $\frac{m v d v}{d t}=K$ or $m v d v=K d t$ Integrating, we get $=\frac{m v^{2}}{2}=K t+c$ Using boundary condition $c=o$ So, $\frac{m v^{2}}{2}=K t$ or $v=\sqrt{2 \frac{K t}{m}}$ or $\frac{d s}{d t}=\sqrt{\frac{2 K t}{m}}$ or $d s=\sqrt{\frac{2 K}{m}} \cdot t^{1 / 2} d t$ On integrating, we get $s=\sqrt{\frac{2 K}{m}} \frac{2}{3} t^{3 / 2}$ $\Rightarrow s \propto t^{3 / 2}$