What is kinetic energy? Derive its expression using graphical analysis.

Kinetic energy is the energy possessed by an object due to its motion. The formula for kinetic energy is:

KE=1/2mv²

Where:

• $m$ is the mass of the object,
• $v$ is its velocity.

Graphical Analysis: Consider an object of mass $m$ initially at rest. If a constant force F is applied to the object, the velocity increases. The work done by the force on the object is equal to the change in its kinetic energy.

From Newton’s second law, we know that:

F=ma

where a is acceleration. By using the equation v=u+at (where $u$ is the  initial velocity), we can find that the velocity $v$ at time $t$ is:

v=at

The work done $W$ is the area under the force vs. displacement graph. The force $F$ is constant, so the area under the graph (which represents work) is given by:

W=F⋅d=ma⋅d

Using the equation d=v²−u²/2a​, and substituting u=0, we get:

W=1/2mv²

Thus, the kinetic energy of the object is:

KE=1/2mv²