A student claims that two cars of equal mass, one moving at 40 km/h and another at 80 km/h, have the same kinetic energy if their engines provide the same power. What is the error in this reasoning?

The correct option is: Kinetic energy depends on the square of speed, not just on power supplied.

To understand why this reasoning is correct, let us break down the physics principles involved: kinetic energy and engine power.

First, the kinetic energy ($E_k$) of an object of mass $m$ moving at a speed $v$ is given by the formula:
$E_k=\frac{1}{2}m{v}^2$
This equation shows that kinetic energy is directly proportional to the square of the speed ($v^2$).

If we compare the two cars of equal mass $m$:
- The first car is moving at $v_1=40\text{km/h}$.
- The second car is moving at $v_2=80\text{km/h}$ (which is $2v_1$).

Comparing their kinetic energies:
$E_{k2}=\frac{1}{2}m{v}_2^2=\frac{1}{2}m{\left(2v_1\right)}^2=4\left(\frac{1}{2}m{v}_1^2\right)=4E_{k1}$
Thus, the car moving at 80 km/h has four times the kinetic energy of the car moving at 40 km/h, despite their equal masses.

Second, power ($P$) is the rate at which work is done or energy is transferred over time ($t$):
$P=\frac{dW}{dt}$
While the engines might supply the same rate of energy (power) at a given instant, the kinetic energy of the vehicle is a state function of its current motion (velocity) and depends directly on the square of its speed, not the instantaneous power output of the engine.