Question

The slope of kinetic energy vs displacement curve of a particle in motion is

• A. equal to the acceleration of the particle
• B. inversely proportional to acceleration
• C. directly proportional to acceleration
• D. none of these

Answer 1

The correct option is C

directly proportional to acceleration

Finding the slope of kinetic energy vs displacement curve graph

We know that,

$Slopeofgraph=\frac{kineticenergy}{displacement}$

Kinetic energy of a particle in motion $E_k=\frac{1}{2}m{v}^2$ where m = mass and v = velocity of particle.
Differentiating with respect to x(displacement), we get

$$\begin{gathered} \frac{d{E}_k}{dx}=\frac{1}{2}m\times 2v\frac{dv}{dx} \\ =mv\times \frac{dv}{dt}\times \frac{dt}{dx} \\ =mv\times a\times \frac{1}{v} \\ =ma \\ \therefore slope\propto acceleration \end{gathered}$$

Thus, the slope of the kinetic energy vs displacement curve of a particle in motion is directly proportional to acceleration.

Hence, option (c) is correct.