Question

A self propelled vehicle of mass $m$ whose engine delivers constant power $P$ has an acceleration $a=\frac{P}{mv}$ (Assume that there is no friction). In order to increase its velocity from $v_1$ to $v_2$, the distance it has to travel will be:

• A. $\frac{3P}{m}(v_2^2-v_1^2)$
• B. $\frac{m}{3p}(v_2^3-v_1^3)$
• C. $\frac{m}{3p}(v_2^2-v_1^2)$
• D. $\frac{m}{3p}(v_2-v_1)$

Answer 1

Answer: (B)

$\frac{m}{3p}(v_2^3-v_1^3)$

Power is given by,
$P=Fv$
$P=mava=\frac{P}{mv}\Rightarrow v\frac{dv}{ds}=\frac{P}{mv}$
$v^{2}dv=\frac{P}{m}ds$

$\Rightarrow \frac{P}{m}{\int }_0^{s}ds={\int }_{v_1}^{v_2}{v}^{2}dv$

$\frac{P}{m}\left(s\right)=\frac{1}{3}(v_2^3-v_1^3)$

$\Rightarrow s=\frac{m}{3P}(v_2^3-v_1^3)$