Question

A self-propelled vehicle of mass m whose engine delivers a constant power P has an acceleration $a=\frac{P}{mv}$. What is the distance travelled by it (assuming no friction) to increase the velocity of the vehicle from $v_1$ to $v_2$?

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

Answer 1

The correct option is B $x=\frac{m}{3P}(v_2^3-v_1^3)$

$a=\frac{dv}{dt}=\frac{dv}{dt}\frac{dx}{dx}=v\frac{dv}{dx}$

$a=\frac{p}{mv}$

So,

$v^{2}dv=\frac{p}{m}dx{\int }_{v_1}^{v_2}{v}^{2}dv={\int }_{x_1}^{x_2}\frac{p}{m}dx{\left[\frac{v^3}{3}\right]}_{v_1}^{v_2}{\frac{p}{m}\left[x\right]}_{x_1}^{x_2}\frac{v_2^3-v_1^3}{3}=\left(p/m\right)(x_2-x_1)(x_2-x_1)=X=\frac{m{(v}_2^3-v_1^3)}{3p}$