The kinetic energy of a moving body is directly proportional to its mass and the square of its speed. If the mass of a bullet is halved, by what factor must its speed be increased to have the same kinetic energy as before?

To determine the factor by which the speed of the bullet must be increased, we need to understand how kinetic energy, mass, and speed are related.

Given that the kinetic energy (KE) of a moving body is directly proportional to its mass (m) and the square of its speed (v), we can write the equation as:

KE ∝ m * v^2

Now, let's denote the initial mass of the bullet as m1, the initial speed as v1, and the final mass as m2. Since the mass of the bullet is halved, we have m2 = m1/2.

To find the factor by which the speed must be increased, we need to determine the ratio of the new speed (v2) to the initial speed (v1):

(v2 / v1)^2 = (m2 / m1) [Using the equation KE ∝ m * v^2]

Plugging in the values, we get:

(v2 / v1)^2 = (m1/2) / m1

(v2 / v1)^2 = 1/2

Taking the square root of both sides:

v2 / v1 = 1/√2 = √2/2

Finally, to find the factor by which the speed must be increased, we divide v2 by v1:

v2 / v1 = (√2/2) / 1 = √2/2

Therefore, the speed of the bullet must be increased by a factor of √2/2 to have the same kinetic energy as before when the mass is halved.