Notes : Kinetic Energy Definition, Properties, Vector Form , Units and Dimensions
Kinetic energy is one of the most important forms of mechanical energy. It is the energy possessed by a body due to its motion. Any moving object, whether it is a car, a rolling ball, or a flying airplane, possesses kinetic energy.
1. Definition of Kinetic Energy
Kinetic Energy (K): The energy possessed by a body by virtue of its motion is called kinetic energy.
It is equal to the amount of work done in bringing a body from rest to a given velocity.
Formula:
\[ K=\frac{1}{2}mv^2 \]
Where:
- m = Mass of the body
- v = Velocity of the body
2. Important Properties of Kinetic Energy
- Kinetic energy is a scalar quantity.
- It depends on both mass and velocity.
- Kinetic energy is always positive or zero.
- A body at rest has zero kinetic energy.
- Kinetic energy depends upon the frame of reference of the observer.
- Kinetic energy is directly proportional to mass.
- Kinetic energy is directly proportional to the square of velocity.
3. Vector Form of Kinetic Energy
Although velocity is a vector quantity, kinetic energy is a scalar quantity.
Using vector notation:
\[ K=\frac{1}{2}m(\vec{v}\cdot\vec{v}) \]
Since,
\[ \vec{v}\cdot\vec{v}=v^2 \]
Therefore,
\[ K=\frac{1}{2}mv^2 \]
4. Derivation of Kinetic Energy
Step 1: Consider a body of mass m
Let a force F act on a body of mass m and increase its velocity from u to v through a displacement s.
According to Newton's Second Law:
\[ F=ma \]
Work done by the force is:
\[ W=Fs \]
Substituting F = ma:
\[ W=mas \]
Step 2: Use the Equation of Motion
We know that:
\[ v^2-u^2=2as \]
Therefore,
\[ as=\frac{v^2-u^2}{2} \]
Step 3: Substitute in Work Equation
\[ W=m\left(\frac{v^2-u^2}{2}\right) \]
\[ W=\frac{1}{2}m(v^2-u^2) \]
The work done equals the change in kinetic energy.
\[ K_f-K_i=\frac{1}{2}m(v^2-u^2) \]
If the body starts from rest (u = 0), then:
\[ K=\frac{1}{2}mv^2 \]
This is the expression for kinetic energy.
5. Calculus Derivation of Kinetic Energy
Let a variable force act on a body of mass m.
The small work done during a very small displacement is:
\[ dW = F\,dx \]
According to Newton's Second Law:
\[ F = ma \]
and
\[ a=\frac{dv}{dt} \]
Therefore,
\[ dW = m\frac{dv}{dt}dx \]
We know that velocity is:
\[ v=\frac{dx}{dt} \]
or
\[ dx=v\,dt \]
Substituting \(dx=v\,dt\) into the work equation:
\[ dW=m\frac{dv}{dt}(v\,dt) \]
Cancelling \(dt\),
\[ dW=mv\,dv \]
Now integrate both sides from initial velocity \(u\) to final velocity \(v\):
\[ \int dW=\int_u^v mv\,dv \]
Since mass \(m\) is constant,
\[ W=m\int_u^v v\,dv \]
Using the integration formula:
\[ \int x\,dx=\frac{x^2}{2} \]
we get
\[ W=m\left[\frac{v^2}{2}\right]_u^v \]
Applying the limits:
\[ W=m\left(\frac{v^2}{2}-\frac{u^2}{2}\right) \]
\[ W=\frac{1}{2}m(v^2-u^2) \]
According to the Work-Energy Theorem,
\[ W=K_f-K_i \]
Therefore,
\[ K_f-K_i=\frac{1}{2}m(v^2-u^2) \]
If the body starts from rest, then
\[ u=0 \]
Hence,
\[ K=\frac{1}{2}mv^2 \]
Final Result
\[ \boxed{K=\frac{1}{2}mv^2} \]
Key Idea
The most important step is:
\[ dW=mv\,dv \]
Integrating \(v\,dv\) gives:
\[ \int v\,dv=\frac{v^2}{2} \]
which directly leads to the kinetic energy formula:
\[ \boxed{K=\frac{1}{2}mv^2} \]
Note: This is the simplest calculus-based derivation of kinetic energy and provides a deeper understanding of how the formula \(K=\frac{1}{2}mv^2\) is obtained.
6. Relation Between Kinetic Energy and Momentum
Linear momentum is:
\[ p=mv \]
Substituting in the kinetic energy formula:
\[ K=\frac{1}{2}mv^2 \]
\[ K=\frac{(mv)^2}{2m} \]
\[ K=\frac{p^2}{2m} \]
Therefore,
\[ K=\frac{p^2}{2m} \]
or
\[ p=\sqrt{2mK} \]
7. Dependence of Kinetic Energy on Mass and Velocity
(a) Dependence on Mass
\[ K\propto m \]
If mass doubles, kinetic energy also doubles.
(b) Dependence on Velocity
\[ K\propto v^2 \]
If velocity doubles, kinetic energy becomes four times.
If velocity becomes three times, kinetic energy becomes nine times.
8. Units and Dimensions of Kinetic Energy
SI Unit
\[ 1~J=1~kg~m^2s^{-2} \]
The SI unit of kinetic energy is Joule (J).
Dimensional Formula
\[ [M^1L^2T^{-2}] \]
9. Examples of Kinetic Energy
- A moving car possesses kinetic energy.
- A flying airplane possesses kinetic energy.
- A rolling ball possesses kinetic energy.
- A moving train possesses kinetic energy.
- A running athlete possesses kinetic energy.
10. FAQs
What is kinetic energy?
Kinetic energy is the energy possessed by a body due to its motion.
What is the formula of kinetic energy?
\[ K=\frac{1}{2}mv^2 \]
Is kinetic energy a scalar quantity?
Yes, kinetic energy is a scalar quantity.
Can kinetic energy be negative?
No. Kinetic energy is always positive or zero.
What is the SI unit of kinetic energy?
The SI unit of kinetic energy is Joule (J).
What is the relation between momentum and kinetic energy?
\[ K=\frac{p^2}{2m} \]
11. MCQs
1. The kinetic energy of a body depends upon:
A) Mass only
B) Velocity only
C) Mass and velocity
D) Shape of body
Answer: C) Mass and velocity
2. The SI unit of kinetic energy is:
A) Newton
B) Joule
C) Watt
D) Pascal
Answer: B) Joule
3. Kinetic energy is:
A) Vector quantity
B) Scalar quantity
C) Neither scalar nor vector
D) Tensor quantity
Answer: B) Scalar quantity
4. If the velocity of a body becomes twice, its kinetic energy becomes:
A) Two times
B) Three times
C) Four times
D) Eight times
Answer: C) Four times
5. A body at rest has kinetic energy:
A) Infinite
B) Negative
C) Zero
D) Positive
Answer: C) Zero
6. The dimensional formula of kinetic energy is:
A) [MLT⁻²]
B) [ML²T⁻¹]
C) [ML²T⁻²]
D) [M²LT⁻²]
Answer: C) [ML²T⁻²]
7. The relation between kinetic energy and momentum is:
A) K = p/2m
B) K = p²/2m
C) K = pm
D) K = m/2p
Answer: B) K = p²/2m
8. A body at rest possesses:
A) Maximum kinetic energy
B) Zero kinetic energy
C) Negative kinetic energy
D) Infinite kinetic energy
Answer: B) Zero kinetic energy
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