Question

The slope of Kinetic Energy $-$ displacement curve of a particle in motion is:

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

Answer 1

The correct option is C directly proportional to the acceleration

using the equation for kinetic energy we have,
$KE=\frac{1}{2}\left(m{v}^2\right)$
$\Delta$$KE$ $=$ $\frac{1}{2}\left(m\right(\Delta v^2\left)\right)$
And, $\Delta v^2=(V_1{)}^2-(V_2{)}^2=(V_1-V_2)\times (V_1+V_2)=\left(\Delta V\right)\times 2\times V$
So, $\Delta$ KE $=$ $m\times \Delta V\times V$
Now,
$V=\frac{\Delta X}{\Delta t}$
$⟹$$\Delta$$KE$ $=$ $\Delta V\times \left(\frac{\Delta X}{\Delta t}\right)\times m$
$⟹$$\left(\frac{\Delta KE}{\Delta X}\right)=m\times \left(\frac{\Delta V}{\Delta t}\right)$
$⟹$$\left(\frac{\Delta KE}{\Delta X}\right)=m\times a$, where $a$ is acceleration.

So, the slope is directly proportional to the acceleration.