Question

The slope of the Kinetic Energy versus Position vector curve gives the rate of change of:

• A. Momentum
• B. Velocity
• C. Force
• D. Power

Answer 1

The correct option is A

Momentum

The explanation for the correct answer:

Correct Answer: (A) Momentum

Let the kinetic energy be $KE$ and the position vector be $x$.

The slope of kinetic energy and position vector curve is given by:

Slope= $\left(\frac{dKE}{\mathrm{dx}}\right)$

Kinetic energy can further be dissolved as: $\mathrm{KE}=\frac{1}{2}{\mathrm{mv}}^2$

$m$is the mass

$v$ is the velocity

From equation (i)

Slope= $\mathrm{mv}\left(\frac{\mathrm{dv}}{\mathrm{dx}}\right)$

Multiplying the numerator and denominator by:

Slope=$\mathrm{mv}\left(\frac{\mathrm{dv}}{\mathrm{dx}}\right)\times \left(\frac{\mathrm{dt}}{\mathrm{dt}}\right)$

$$\begin{gathered} =\mathrm{mv}\left(\frac{\mathrm{dv}}{\mathrm{dt}}\right)\times \left(\frac{\mathrm{dt}}{\mathrm{dx}}\right) \\ =\mathrm{mv}\left(\mathrm{a}\right)\times \frac{1}{\mathrm{v}}\left\{\left(\frac{\mathrm{dv}}{\mathrm{dx}}\right)=\mathrm{a},\mathrm{and}\left(\frac{\mathrm{dt}}{\mathrm{dx}}\right)=\mathrm{v}\right\} \\ =\mathrm{ma} \end{gathered}$$

$a$ is the acceleration

Also, we know that $F=ma$

According to Newton's second law of laws of motion, the rate of change of momentum is called force.

$F=\frac{dp}{dt}$

Therefore, the slope of the Kinetic Energy versus Position vector curve gives the rate of change of momentum.

The explanations for incorrect options:

Option (C) Force

1. As we have proved above the slope of the Kinetic Energy versus Position vector curve gives force and not the rate of change of force.
2. Therefore, this option is incorrect.

Option (B) Velocity

1. Velocity is obtained by the slope of the displacement-time graph.
2. The slope of the Kinetic Energy versus Position vector curve does not give the rate of change of velocity.
3. Therefore, this option is incorrect.

Option (D) Power

1. Power is defined as the work done per unit of time.
2. The slope of the Kinetic Energy versus Position vector curve does not give the rate of change of power.
3. Therefore, this option is incorrect.

Hence, the slope of the Kinetic Energy versus Position vector curve gives the rate of change of momentum.

Therefore, option (A) is the correct answer.