Introduction:
When the linear momentum of a body is increased by a certain percentage, it is natural to wonder how it affects the kinetic energy of the body. To understand this relationship, we need to delve into the concepts of linear momentum and kinetic energy and explore their mathematical connection.

Linear Momentum:
Linear momentum, denoted by "p," is a vector quantity that measures the motion of an object. It is defined as the product of an object's mass (m) and its velocity (v), as shown by the equation: p = m * v. Since momentum depends on both mass and velocity, any change in either of these factors affects the linear momentum of the body.

Kinetic Energy:
Kinetic energy, denoted by "KE," is the energy possessed by an object due to its motion. It is given by the equation: KE = (1/2) * m * v^2, where m represents the mass of the object and v denotes its velocity. Here, the kinetic energy depends on the square of the velocity, making it sensitive to changes in the object's speed.

Relationship between Linear Momentum and Kinetic Energy:
To understand how a change in linear momentum affects kinetic energy, let's consider a scenario where the linear momentum of a body is increased by 10%. We can express this change mathematically as: p' = 1.1 * p, where p' represents the new linear momentum and p represents the original linear momentum.

Deriving the Percentage Change in Kinetic Energy:
To determine the percentage change in kinetic energy resulting from the 10% increase in linear momentum, we compare the initial kinetic energy (KE) to the final kinetic energy (KE'). Using the equations mentioned earlier, we can express these values as follows:

Initial Kinetic Energy: KE = (1/2) * m * v^2
Final Kinetic Energy: KE' = (1/2) * m * (1.1 * v)^2

Simplifying the Equations:
By simplifying the equations, we can calculate the ratio of the final kinetic energy to the initial kinetic energy:

KE'/KE = [(1/2) * m * (1.1 * v)^2] / [(1/2) * m * v^2]
= (1.1^2 * v^2) / v^2
= 1.1^2
= 1.21

Calculating the Percentage Change:
To determine the percentage change in kinetic energy, we subtract 1 from the ratio we obtained and multiply by 100:

Percentage Change = (KE'/KE - 1) * 100
= (1.21 - 1) * 100
= 0.21 * 100
= 21%

Conclusion:
When the linear momentum of a body is increased by 10%, its kinetic energy increases by 21%. This relationship arises due to the quadratic dependence of kinetic energy on velocity. By understanding the mathematical connection between these two quantities, we can gain insights into the behavior of objects in motion and their associated energies.

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