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?
YES
To find the factor by which the speed of the bullet must be increased, we need to understand the relationship between kinetic energy, mass, and speed.
The formula for kinetic energy (KE) is:
KE = (1/2) * mass * speed^2
According to the question, we want to maintain the same kinetic energy after halving the mass of the bullet. Let's assume the initial mass is M and the initial speed is V. So, the initial kinetic energy (KEi) is given by:
KEi = (1/2) * M * V^2
After halving the mass, the new mass (M') becomes (1/2)M. Let's assume the new speed is V'. The new kinetic energy (KEn) is given by:
KEn = (1/2) * (1/2)M * V'^2
Since we want both kinetic energies to be equal, we can equate the initial and new kinetic energy:
(1/2) * M * V^2 = (1/2) * (1/2)M * V'^2
By canceling out the common factors, we get:
V^2 = (1/2) * V'^2
To find the factor by which the speed must be increased, we can take the square root of both sides:
√(V^2) = √((1/2) * V'^2)
V = (1/√2) * V'
To isolate V', we can divide both sides by (1/√2):
V' = √2 * V
Therefore, the speed of the bullet must be increased by a factor of √2 (approximately 1.414) to maintain the same kinetic energy after halving the mass.