The velocity 'v' reached by a car of mass 'm' at certain distance from the starting point driven with contatant power 'P' is such that

Work performed in a given length of time is referred to as mechanical power. We are aware that the amount of work done on an item is calculated as the sum of the force that is applied to it and the displacement that it undergoes as a result of that force. given mathematically as,

$W=F\times s$

$P=\frac{W}{t}=\frac{F\times s}{t}=Fs$

Since the object's velocity is equal to its displacement over time, This calculation tells us that the car travels x distance at speed v in our inquiry. P=Fv can now be used to determine power.
We are aware that an object's force can be expressed as the product of its mass and acceleration. Using this in the equation above, we obtain P=mav.
We are aware that a body's acceleration is the rate of change in velocity per unit of time. Using this in the equation above, we obtain,

$$\begin{gathered} P=m\frac{dv}{dt}\times v \\ \Rightarrow \frac{P}{m}=v\frac{dv}{dt} \\ \Rightarrow \frac{P}{m}=v\frac{dv}{dx}\times \frac{dx}{dt} \\ \Rightarrow \frac{P}{m}=v^2\frac{dv}{dx} \\ \Rightarrow \int \frac{P}{m}dx=\int v^{2}dv \\ \Rightarrow \frac{P}{m}x=\frac{v^3}{3} \\ \Rightarrow v^3=\frac{3Px}{m} \\ \Rightarrow v={\left(\frac{3xP}{m}\right)}^{\frac{1}{3}} \\ \Rightarrow v^3\propto \frac{3p}{m} \end{gathered}$$

Hence the correct answer is $v^3\propto \frac{3p}{m}.$