Question

A self – propelled vehicle of mass m, whose engine delivers a constant power P, has an acceleration a = (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. $\frac{m}{3P}(v_2-v_1{)}^3$
• B. $\frac{m}{3P}(v_2^2-v_1^2)$
• C. $\frac{m}{3P}(v_2^3-v_1^3)$
• D. $\frac{3P}{m}(v_2^2-v_1^2)$

Answer 1

The correct option is C

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

$=\frac{vdv}{dx}=\frac{P}{mv}$

${\int }_{v_1}^{v_2}{v}^{2}dv={\int }_0^{x}Pdx$

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