the kinetic energy of moving body ids directly praportional 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?
E = k * m * v^2
If we halve the mass, now we have
E' = k * m/2 * v^2 = E/2
In order to get E' = E, we need a new velocity, v'
E' = k * m/2 * v'^2
and we see that
v'^2 must be 2v^2
so, v' = v√2
To determine the factor by which the speed of the bullet must be increased to maintain the same kinetic energy, we need to analyze the relationship between kinetic energy (KE), mass (m), and speed (v).
The formula for kinetic energy is KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity (speed).
Given that kinetic energy is directly proportional to the square of the speed, we can express it as KE ∝ v^2. If we express this proportionality as an equation, we get KE = k * v^2, where k is a constant.
Now, let's solve the problem:
1. Let's assume the initial mass of the bullet is 'm' and its initial speed is 'v'.
2. The initial kinetic energy (KE1) can be calculated using KE = (1/2)mv^2.
3. Now, the mass is halved, so the new mass of the bullet is (m/2).
4. We want to find the factor by which the speed must be increased, so let's assume the new speed is 'xv', where 'x' is the factor we need to determine.
5. The new kinetic energy (KE2) can be calculated using KE = (1/2)(m/2)(xv)^2.
6. Since we want KE2 to be equal to KE1, we can set up the equation: (1/2)mv^2 = (1/2)(m/2)(xv)^2.
7. Simplifying this equation, we get mv^2 = (m/2)(x^2)(v^2).
8. Canceling out the v^2 terms, we have m = (m/2)(x^2).
9. Dividing both sides of the equation by (m/2), we get 1 = x^2/2.
10. Multiply both sides by 2, and we have 2 = x^2.
11. Taking the square root of both sides, we find that x = √2.
Therefore, the factor by which the speed of the bullet must be increased to have the same kinetic energy as before is approximately √2, which is approximately 1.41.