Derivation of Kinetic Energy - Definition, Equation, Example, FAQs

Definition of Kinetic Energy

Kinetic Energy: The kinetic energy of a moving body is the measure of work required to bring the body from rest to its present state of motion, or the work it can perform before coming to rest.

A body possesses kinetic energy due to its motion, and this energy depends on both the mass of the body and its velocity.

Derivation of Kinetic Energy (Using Constant Force)

Consider a body of mass $m$ initially at rest.
A constant force $\mathbf{F}$ acts on it, producing an acceleration a.

From Newton's Second Law:

$
a=\frac{F}{m}
$
Let the body attain a velocity $\mathbf{v}$ after moving through a distance $\mathbf{s}$.
Using the kinematic equation:

$
\begin{gathered}
v^2=2 a s=2\left(\frac{F}{m}\right) s \\
F \cdot s=\frac{1}{2} m v^2
\end{gathered}
$

The work done,

$
W=F \cdot s
$

This work is stored as kinetic energy $(\mathrm{K})$, therefore:

$
K=\frac{1}{2} m v^2
$
Kinetic Energy $=\frac{1}{2} \times$ mass $\times$ velocity ${ }^2$
This shows speed has a greater effect on kinetic energy compared to mass because velocity is squared.

• Unit: Joule
• Dimensional Formula: $\left[M L^2 T^{-2}\right]$

General Derivation of Kinetic Energy (Variable Force)

If a variable force $\mathbf{F}$ acts on a body of mass $\mathbf{m}$, the infinitesimal work done for a displacement ds is:

$
d W=\vec{F} \cdot d \vec{s}
$

Using Newton's law:

$
F=m a=m \frac{d v}{d t}
$

Substituting:

$
d W=\left(m \frac{d v}{d t}\right) d s=m v d v \quad\left(\because v=\frac{d s}{d t}\right)
$

Total work done from velocity 0 to v :

$
\begin{gathered}
W=\int_0^v m v d v \\
W=\frac{1}{2} m v^2
\end{gathered}
$
Thus,

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

Kinetic energy is a scalar quantity.

Derivation of Kinetic Energy (Class 9 Method)

A body of mass $\mathbf{m}$ moving with velocity $\mathbf{v}$ is brought to rest by a retarding force $\mathbf{F}$ over a distance $\mathbf{s}$.

Initial velocity $=\mathrm{v}$
Final velocity $=0$

Using kinematics:

$
0=v^2+2 a s
$

Since $\mathbf{a}$ is negative:

$
s=\frac{v^2}{2 a}
$

Work done by retarding force:

$
\begin{gathered}
W=F \cdot s=(m a) \cdot \frac{v^2}{2 a} \\
K E=\frac{1}{2} m v^2
\end{gathered}
$

Work-Energy Theorem

The work-energy theorem states:
The work done by a force on a body equals the change in the kinetic energy of the body.

If a body's velocity changes from $\mathbf{u}$ to $\mathbf{v}_{\text {, then: }}$

$
\begin{gathered}
W=\frac{1}{2} m v^2-\frac{1}{2} m u^2 \\
W=\Delta K
\end{gathered}
$

Thus,

$
W=\Delta K
$

Types of Kinetic Energy

• Thermal energy
• Radiant energy
• Sound energy
• Mechanical energy
• Electrical energy

All these are forms of kinetic energy.

Potential Energy
Potential energy is the energy possessed by a body due to its position or state of deformation.

Example: Water at a height possesses potential energy which can be used to rotate a turbine.

Also Read:

• NCERT solutions for Class 11 Physics Chapter 6 Work, Energy and Power
• NCERT Exemplar Class 11 Physics Solutions Chapter 6 Work, Energy and Power
• NCERT notes Class 11 Physics Chapter 6 Work, Energy and Power

Potential Energy

Potential energy is the energy possessed by a body due to its position or state of deformation.

Example: Water at a height possesses potential energy which can be used to rotate a turbine.

Gravitational Potential Energy

If a body of mass $m$ is raised through a height $h$, work must be done against gravity.

$
W=m g h
$

This work is stored as:

$
U=m g h
$

If the body falls back, it can give back the same amount of work.

Also check-

• NCERT Exemplar Class 11th Physics Solutions
• NCERT Exemplar Class 12th Physics Solutions
• NCERT Exemplar Solutions for All Subjects

NCERT Physics Notes:

• NCERT Notes Class 11th Physics
• NCERT Notes Class 12th Physics
• NCERT Notes For All Subjects