Momentum and Kinetic Energy:
Momentum and kinetic energy are two important concepts in physics that describe the motion of an object. Momentum is a measure of an object's motion, while kinetic energy is a measure of an object's energy due to its motion.

Relationship between Momentum and Kinetic Energy:
The relationship between momentum and kinetic energy can be understood by examining their definitions. Momentum is defined as the product of an object's mass and its velocity, while kinetic energy is defined as one-half the product of an object's mass and the square of its velocity.

Mathematical Expressions:
The mathematical expressions for momentum (p) and kinetic energy (K) are as follows:
- Momentum (p) = mass (m) × velocity (v)
- Kinetic Energy (K) = 1/2 × mass (m) × velocity squared (v²)

Percentage Increase in Momentum:
Let's assume the initial momentum of the object is P, and its final momentum after a 5% increase is P'. The percentage increase in momentum can be calculated using the formula:
- Percentage Increase in Momentum = (P' - P) / P × 100

Percentage Increase in Kinetic Energy:
To find the percentage increase in kinetic energy, we need to determine the initial kinetic energy (K) and the final kinetic energy (K').

Initial Kinetic Energy:
The initial kinetic energy can be calculated using the formula:
- K = 1/2 × mass (m) × initial velocity squared (v²)

Final Kinetic Energy:
The final kinetic energy can be calculated using the formula:
- K' = 1/2 × mass (m) × final velocity squared (v'²)

Relationship between Velocity and Momentum:
Since momentum is directly proportional to velocity, we can say that the change in momentum is directly proportional to the change in velocity.

Deriving the Relationship:
- Initial Momentum (P) = mass (m) × initial velocity (v)
- Final Momentum (P') = mass (m) × final velocity (v')

Percentage Increase in Momentum:
Using the formula for percentage increase in momentum, we have:
- Percentage Increase in Momentum = (P' - P) / P × 100
- Percentage Increase in Momentum = (mass (m) × final velocity (v') - mass (m) × initial velocity (v)) / (mass (m) × initial velocity (v)) × 100
- Percentage Increase in Momentum = (v' - v) / v × 100

Deriving the Relationship:
- Initial Kinetic Energy (K) = 1/2 × mass (m) × initial velocity squared (v²)
- Final Kinetic Energy (K') = 1/2 × mass (m) × final velocity squared (v'²)

Percentage Increase in Kinetic Energy:
Using the formula for percentage increase in kinetic energy, we have:
- Percentage Increase in Kinetic Energy = (K' - K) / K × 100
- Percentage Increase in Kinetic Energy = ((1/2 × mass (m) × final velocity squared (v'²)) - (1/2 × mass (m) × initial velocity squared (v²

K.E=(200+x)x/100
where x is % of momentum increased
so K.E=(200+5)5/100
K.E=(205)5/100
K.E=205/20
K.E=10.5%