Kinetic Energy: Definition, Formula, Derivation, and More

The kinetic energy of an object is a measure of the work an object can do by the virtue of its motion. If an object of mass m has velocity v, its kinetic energy K is

$K=\frac{1}{2}m{v}^2=\frac{1}{2}m\vec{v}\cdot \vec{v}$

Where  $v=\left|\vec{v}\right|$ is the velocity of the object.

And m = mass of the object.

Kinetic energy is a scalar quantity and can also be expressed in terms of linear momentum  $(\vec{p}=m\vec{v})$ as:

$K=\frac{p^2}{2m}$

*Note: Kinetic Energy can never be a negative value since both mass and velocity are positive integers.

In simple terms, Kinetic energy is referred to the energy that a body possesses due to its movement. For e.g.: a ball rolling down the hill, a moving car, running athletes etc., are all examples of kinetic energy. SI unit for Kinetic Energy is measured in terms of Joule (J).

The Kinetic energy of a body mainly depends on two major factors: speed of the body and mass of the body.

Here are some of the core aspects of Kinetic Energy which the JEE aspirants can refer to:

• Kinetic Energy can also be converted from one form to another, such as potential energy.
• Kinetic Energy is a scalar quantity i.e. it doesn’t have any directions and only has a magnitude.
• Kinetic Energy can never be a negative value (since both mass and velocity are positive integers).
• If the velocity of a moving body is doubled, the Kinetic Energy of the body will increase 4 times (K.E. = ½ * m * v^2)
• At the same speed, objects with higher mass will possess more Kinetic Energy as compared to those with comparatively less weight.

The Work Energy theorem is a crucial part of physics which states that the sum of all forces applied on a body will equal to the change in its Kinetic Energy.

Mathematical representation:

$W_{\mathrm{n}\mathrm{e}\mathrm{t}}=\mathrm{\Delta }K=K_f-K_i=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

For a particle moving from point  $P_1$ to  $P_2$ under a net force  $\vec{F}$ , the work done is:

$W={\int }_{P_1}^{P_2} \vec{F}\cdot d\vec{r}$

The force applied on an object will affect its motion, resulting in a change in the kinetic energy of the particular object.

Here’s a step by step explained derivation of Kinetic Energy formula using calculus:

W=∫Fdx (work done by a force)

F=ma(Newton’s second law of motion)

a=dt/dv​
So, a=dtdv​=dxdv​⋅dtdx​=dxdv​⋅v

F=ma=m⋅v⋅dxdv​
Now, use the work formula, i.e.:

W=∫Fdx=∫mvdxdv​dx

W=∫mvdv

W=m∫vdv=m⋅2v2​+C

If the object starts from zero (rest), both v and c will be zero.

Therefore,
KE=1/2*m*v^2

Hence, proved.

Kinetic energy can be majorly classified into three categories:

• Translational Kinetic energy: This the most common form of kinetic energy which the body possesses due to its’ motion from one point to another in a straight line. Eg: A car moving on a road.
• Rotational Kinetic energy: This is the energy possessed by the object due to its’ rotation around an axis. Eg: A Beyblade.
• Vibrational Kinetic energy: This is the energy possessed by the body due to its’ vibration. Eg: a vibrating mobile phone.

There is a basic difference in both the terms kinetic energy and potential energy in the way where they are used and how can they be combined to form mechanical energy. these differences are explained in the simplest terms as follows:

Kinetic Energy:

• Kinetic Energy is the energy possessed by the body due to its’ motion.
• Example: a ball rolling down the hill.
• Formula: ½ * m * v^2

Potential energy:

• Potential Energy is the energy possessed by the body due to its’ position.
• Example: a stretched rubber band.
• Formula: mgh