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?
sqrt2.
To determine the factor by which the speed must be increased if the mass of a bullet is halved and its kinetic energy remains the same, let's start by understanding the relationship between kinetic energy, mass, and speed.
According to the given information, the kinetic energy (KE) of a moving body is directly proportional to its mass (m) and the square of its speed (v). Mathematically, this can be expressed as:
KE ∝ m * v^2
Now, let's consider the situation when the mass of the bullet is halved. We can represent the original mass as m1 and the halved mass as m2. So, m2 = m1/2.
To maintain the same kinetic energy (KE) as before, we need to find the factor by which the speed (v2) must be increased. Let's calculate it step by step.
1. Original kinetic energy: KE1 = m1 * v1^2
2. New kinetic energy: KE2 = m2 * v2^2
Since kinetic energy remains the same, we can equate the two equations:
m1 * v1^2 = m2 * v2^2
Substituting the value of m2: m1 * v1^2 = (m1/2) * v2^2
Next, we can cancel out the common factor (m1) from both sides:
v1^2 = (1/2) * v2^2
To solve for v2, we can take the square root of both sides:
v1 = (1/√2) * v2
Finally, rearranging the equation to isolate v2, we get:
v2 = (√2) * v1
Therefore, the speed of the bullet must be increased by a factor of √2 (approximately 1.414) to maintain the same kinetic energy when the mass is halved.