The relationship between kinetic energy and momentum can be understood through the equation:

Momentum = Mass × Velocity
Kinetic Energy = 0.5 × Mass × Velocity^2

Let's assume the initial kinetic energy (KE) of a body is K, and its initial momentum is P.

Initial Kinetic Energy: K
Initial Momentum: P

1. Increasing the Kinetic Energy by 300%
To increase the kinetic energy by 300%, we multiply the initial kinetic energy by 3.

New Kinetic Energy: 3K

2. Relationship between Kinetic Energy and Velocity
Since kinetic energy is proportional to the square of velocity, we can write:

K = 0.5 × Mass × Velocity^2

Taking the square root of both sides:

√(2K/mass) = Velocity

3. Relationship between Velocity and Momentum
We know that momentum is equal to mass multiplied by velocity:

P = Mass × Velocity

4. Increasing the Momentum
Now, let's calculate the new momentum using the increased kinetic energy.

New Velocity = √(2 × 3K/mass) = √(6K/mass)
New Momentum = Mass × New Velocity = Mass × √(6K/mass) = √(6K × Mass)

5. Ratio of New Momentum to Initial Momentum
The ratio of the new momentum to the initial momentum can be calculated as follows:

(New Momentum / Initial Momentum) = (√(6K × Mass) / P)
= √(6K × Mass) / (Mass × Velocity)
= √(6K × Mass) / (√(2K × Mass))
= √(6K / 2K)
= √3

This means that the momentum will increase by a factor of √3 or approximately 1.732.

6. Percentage Increase in Momentum
To calculate the percentage increase, we subtract 1 from the ratio and multiply by 100:

Percentage Increase = ((√3 - 1) / 1) × 100
≈ 73.2%

Therefore, the momentum will increase by approximately 73.2%.