K.E. of a body is increased by 44%. What is the percentage increase in the momentum?A) 10% B) 20% C) 30% D) 44%

The kinetic energy (K.E.) of a body is given by the formula:

K.E. = (1/2) * m * v^2

where:
K.E. is the kinetic energy
m is the mass of the body
v is the velocity of the body

Now, let's consider the initial kinetic energy as K.E₁ and the final kinetic energy as K.E₂.

If the kinetic energy is increased by 44%, we can write this as:

K.E₂ = K.E₁ + 0.44 * K.E₁
K.E₂ = 1.44 * K.E₁

Now, let's consider the initial momentum as p₁ and the final momentum as p₂.

The momentum (p) of a body is given by the formula:

p = m * v

Now, we want to find the percentage increase in momentum, which can be calculated as follows:

Percentage Increase in Momentum = [(p₂ - p₁) / p₁] * 100

Substitute the expressions for momentum (p) in terms of mass and velocity:

Percentage Increase in Momentum = [(m * v₂ - m * v₁) / (m * v₁)] * 100

Notice that the mass (m) is common in both the initial and final momentum, so it cancels out when calculating the percentage increase. Therefore, we have:

Percentage Increase in Momentum = [(v₂ - v₁) / v₁] * 100

Now, let's use the relationship between kinetic energy and velocity:

K.E = (1/2) * m * v^2

So, we can write:

K.E₁ = (1/2) * m * v₁^2
K.E₂ = (1/2) * m * v₂^2

Now, let's express v₁ and v₂ in terms of K.E₁ and K.E₂:

v₁ = sqrt((2 * K.E₁) / m)
v₂ = sqrt((2 * K.E₂) / m)

Substitute these expressions back into the percentage increase in momentum equation:

Percentage Increase in Momentum = [(sqrt((2 * K.E₂) / m) - sqrt((2 * K.E₁) / m)) / sqrt((2 * K.E₁) / m)] * 100

Now, substitute K.E₂ = 1.44 * K.E₁:

Percentage Increase in Momentum = [(sqrt((2 * 1.44 * K.E₁) / m) - sqrt((2 * K.E₁) / m)) / sqrt((2 * K.E₁) / m)] * 100

Percentage Increase in Momentum = [(sqrt(2.88 * (K.E₁ / m)) - sqrt(2 * (K.E₁ / m))) / sqrt(2 * (K.E₁ / m))] * 100

Percentage Increase in Momentum = [(sqrt(2.88) - sqrt(2)) / sqrt(2)] * 100

Now, calculate this expression:

Percentage Increase in Momentum ≈ 30.86%

So, the percentage increase in momentum is approximately 30.86%, which is closest to option C) 30%.