Kinetic Energy and Momentum

Kinetic energy and momentum are two important concepts in physics that describe the motion of objects. Kinetic energy is the energy an object possesses due to its motion, while momentum is a measure of the object's motion and is defined as the product of its mass and velocity.

Given Information

In this question, we are given that the kinetic energy of a body increases by 2.5%. We need to find the percentage increase in momentum.

Relationship between Kinetic Energy and Momentum

To solve this problem, we need to understand the relationship between kinetic energy and momentum. The kinetic energy of an object can be expressed in terms of its momentum using the equation:

Kinetic Energy = (1/2) * m * v^2

Where m is the mass of the object and v is its velocity.

Calculating the Percentage Increase in Kinetic Energy

Let's assume the initial kinetic energy of the body is KE1 and the final kinetic energy after the increase is KE2.

The percentage increase in kinetic energy can be calculated using the following formula:

Percentage Increase = ((KE2 - KE1) / KE1) * 100

Given that the kinetic energy increases by 2.5%, we have:

2.5% = ((KE2 - KE1) / KE1) * 100

Simplifying the equation, we get:

0.025 = (KE2 - KE1) / KE1

0.025 * KE1 = KE2 - KE1

1.025 * KE1 = KE2

Therefore, the final kinetic energy (KE2) is 1.025 times the initial kinetic energy (KE1).

Calculating the Percentage Increase in Momentum

Now, we need to calculate the percentage increase in momentum. Let's assume the initial momentum of the body is p1 and the final momentum after the increase is p2.

The momentum of an object can be expressed as:

Momentum = m * v

Where m is the mass of the object and v is its velocity.

Since mass is constant, the percentage increase in momentum can be calculated using the equation:

Percentage Increase = ((p2 - p1) / p1) * 100

Substituting the equation for momentum, we get:

Percentage Increase = ((m * v2 - m * v1) / (m * v1)) * 100

Simplifying the equation, we have:

Percentage Increase = ((v2 - v1) / v1) * 100

Now, since kinetic energy is given by (1/2) * m * v^2, we can write:

v2^2 - v1^2 = (2 * KE2 / m) - (2 * KE1 / m)

Simplifying further, we have:

(v2 - v1)(v2 + v1) = (2 * KE2 / m) - (2 * KE1 / m)

v2 - v1 = [(2 * KE2 / m) - (2 * KE1 / m)] / (v2 + v1)

v2 - v1 = (2 * (KE2 - KE1)) / (m * (v2 + v1))

Now, substituting the value of KE2 / KE1 from the earlier calculation, we have:

v