Question

The speed v reached by a car of mass m in travelling a distance x, driven with constant power P, is given by
A. $v=\dfrac{3xP}{m}$
B. $v={{\left( \dfrac{3xP}{m} \right)}^{\dfrac{1}{2}}}$
C. $v={{\left( \dfrac{3xP}{m} \right)}^{\dfrac{1}{3}}}$
D. $v={{\left( \dfrac{3xP}{m} \right)}^{2}}$

Answer 1

Hint: We could derive the required expression from the expression for power as the product of force and velocity. Now you could substitute for force using Newton’s second law of motion. Then you could further reduce acceleration as the differential of velocity. After making necessary rearrangements you could integrate to get the answer.
Formula used:
Expression for power,
$P=F\times v$

Complete answer:
In the question, we are given a car of mass m that is traveling a distance x with speed v. We are supposed to find which among the given options best describes the motion if this car was driven with constant power P.
In order to answer this question, let us recall the expression for power. We know that power by definition is the product of force and velocity, that is,
$P=F\times v$ ……………………………………… (1)
Where, F is the force on the car and v is its speed.
But, by Newton’s second law we know that,
$F=ma$
Where, m is the mass of the car and $a$ its acceleration.
But, we know that acceleration by definition is the time rate of change of velocity, that is,
$a=\dfrac{dv}{dt}$
$\Rightarrow F=m\dfrac{dv}{dt}$
So, now (1) becomes,
$P=\left( m\dfrac{dv}{dt} \right)\times v$
Multiplying and dividing the right hand side of the equation by$dx$ ,
$P=m\dfrac{dv}{dt}\times \dfrac{dx}{dx}\times v=m\dfrac{dv}{dx}\times \dfrac{dx}{dt}\times v$
But,
$\dfrac{dx}{dt}=v$
$\Rightarrow P=m\dfrac{dv}{dx}\times v\times v$
$\Rightarrow Pdx=m{{v}^{2}}dv$
Integrating on both sides,
$P\int{dx=m\int{{{v}^{2}}}}dv$
$\Rightarrow Px=m\dfrac{{{v}^{3}}}{3}$
$\therefore v={{\left( \dfrac{3Px}{m} \right)}^{\dfrac{1}{3}}}$
Therefore, we found the expression that correctly defines the motion to be,
$v={{\left( \dfrac{3Px}{m} \right)}^{\dfrac{1}{3}}}$

Hence, option C is found to be the correct answer.

Note:
You may have noted that during integration, we have taken certain terms outside of the integral symbol. This is because they are constant throughout the motion. The two quantities are power and mass. We know the mass of the car remains the same for the motion and the power is also said to be constant in the question.