Kinetic Energy and Potential Energy 

1.0What Is Energy?

In physics, energy is the capacity to do work. It's a scalar quantity and can exist in various forms, such as kinetic, potential, thermal, and chemical energy. According to the law of conservation of energy, energy cannot be created or destroyed, only transformed from one form to another. For JEE, understanding the mechanical forms of energy—kinetic and potential—is crucial.

2.0What Is Kinetic Energy?

Kinetic energy (KE) is the energy that an object possesses due to its motion. Any object in motion, regardless of its size, has kinetic energy. The faster an object moves or the greater its mass, the more kinetic energy it has. The energy of a moving car, a flying bird, or a rolling ball are all examples of kinetic energy.

3.0Kinetic Energy Formula

$KE=\frac{1}{2}m{v}^2$
Where:

• (m) = mass of the object (kg)
• (v) = velocity of the object (m/s)

4.0Derivation of Kinetic Energy Equation

Work done to accelerate a body from rest to velocity (v):

$W=F\cdot s=ma\cdot s$
Using kinematics:

$v^2-u^2=2as(u=0)$

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

$W=ma\cdot \frac{v^2}{2a}=\frac{1}{2}m{v}^2$
Thus, work done = kinetic energy gained.

5.0Units of Kinetic Energy

• SI Unit: Joule (J)
• 1 Joule = Work done when a force of 1 Newton displaces an object by 1 meter.

6.0Examples of Kinetic Energy

• A moving car
• Flowing water in a river
• Wind turning turbine blades

7.0What Is Potential Energy?

Potential energy (PE) is the energy stored in an object due to its position, state, or configuration. It's often referred to as "stored energy" that has the potential to be converted into other forms of energy. 

Types of Potential Energy

a) Gravitational Potential Energy

Energy due to the height of an object in a gravitational field.
PE = mgh
Where (m) = mass, (g) = acceleration due to gravity, (h) = height.

b) Elastic Potential Energy

Stored energy in a stretched or compressed spring.

$PE=\frac{1}{2}k{x}^2$
Where (k) = spring constant, (x) = deformation.

c) Electric Potential Energy

Energy due to the position of charges in an electric field.

Examples of Potential Energy

• Water stored in a dam (gravitational PE)
• A compressed spring (elastic PE)
• Charged particles in an electric field

8.0Relation Between Kinetic Energy and Potential Energy

Conversion of Energy (KE ↔ PE)

(K.E + P.E) = Constant

As one form of energy increases, the other decreases, and vice versa.

Energy can shift between kinetic and potential forms. For example, in projectile motion:

• At the highest point, maximum PE and minimum KE.
• At the lowest point, maximum KE and minimum PE.

Law of Conservation of Energy

Total mechanical energy = Kinetic Energy + Potential Energy remains constant in an isolated system.

$E_{total}$ = KE + PE = Constant

Real-Life Applications

• Roller coasters (conversion between KE and PE)
• Pendulum motion
• Hydroelectric power generation

9.0Differences Between Kinetic Energy and Potential Energy

10.0Solved Problems

Problem 1: A ball of mass 0.5 kg is dropped from a height of 10 m. What is its velocity just before it hits the ground? (Assume g=10 m/s^2).

Solution: 

Using the conservation of mechanical energy:

$P{E}_{initial}+K{E}_{initial}=P{E}_{final}+K{E}_{final}$

At the top, v = 0, so $K{E}_{initial}$ = 0. At the bottom, h = 0, so $P{E}_{final}$ = 0.

$mg{h}_{initial}+0=0+\frac{1}{2}m{v}_{final}^2$

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

$v=\sqrt{2gh}=\sqrt{2\times 10\times 10}=\sqrt{200}=10\sqrt{2}\text{ m/s}$

Problem 2: A block of mass 2 kg compresses a spring with a spring constant of 200 N/m by 10 cm. What is the maximum speed of the block when the spring is released?

Solution: Energy is converted from elastic potential energy to kinetic energy. 

$P{E}_{initial}=K{E}_{final}$

$\frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2$

$k{x}^2=m{v}^2$

$(200)(0.1{)}^2=(2)v^2$

$2=2v^2$

$v^2=1⟹v=1\text{ m/s}$