Kinetic Energy and Momentum

The kinetic energy (KE) of a particle is given by the equation KE = 1/2 mv^2, where m is the mass of the particle and v is its velocity. The momentum (p) of a particle is given by the equation p = mv, where m is the mass of the particle and v is its velocity.

Given Information:
Kinetic energy of a particle is increased by 300%.

Calculating the % Increase in Momentum:

To determine the % increase in momentum, we need to compare the initial momentum (p1) with the final momentum (p2) of the particle.

Step 1: Assign values to the initial kinetic energy (KE1) and final kinetic energy (KE2) based on the given information.

Let's assume the initial kinetic energy is KE1, and the final kinetic energy is KE2. Since the kinetic energy is increased by 300%, we have KE2 = KE1 + 300% of KE1.

Step 2: Convert the % increase in kinetic energy to a decimal.

To convert a percentage to a decimal, divide the percentage by 100. In this case, 300% is equal to 3 in decimal form (300/100 = 3).

Step 3: Use the equation for kinetic energy to express the relationship between KE1 and KE2.

KE2 = KE1 + 3KE1

Simplifying the equation, we get:

KE2 = 4KE1

Step 4: Express the relationship between momentum and kinetic energy.

Since momentum (p) is equal to the product of mass (m) and velocity (v), we can rewrite the equation for kinetic energy in terms of momentum:

KE = 1/2 mv^2
KE = 1/2 (m)(v^2)
KE = 1/2 (m)(v)(v)
KE = (1/2)(m)(v)

Thus, KE is directly proportional to momentum.

Step 5: Calculate the initial momentum (p1) and final momentum (p2) using the equations for kinetic energy and momentum.

For the initial momentum (p1):
p1 = (1/2)(m)(v1)

For the final momentum (p2):
p2 = (1/2)(m)(v2)

Step 6: Find the relationship between v1 and v2 based on the relationship between KE1 and KE2.

Since KE2 = 4KE1, we can express the relationship between v1 and v2 as:

(1/2)(m)(v2)^2 = 4[(1/2)(m)(v1)^2]
(v2)^2 = 4(v1)^2
v2 = 2v1

Step 7: Substitute the relationship between v1 and v2 into the equations for p1 and p2.

For p1:
p1 = (1/2)(m)(v1)

For p2:
p2 = (1/2)(m)(2v1)
p2 = (1/2)(m)(2)(v1)
p2 = 2[(1/2)(m)(v1)]
p2 =