If a sprinter’s velocity increases to three times the original velocity, by what factor does the kinetic energy increase?
nine.
Kinetic energy is proportional to V^2
The kinetic energy will increase if velocity increases?
The kinetic energy of an object is given by the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Let's assume the initial kinetic energy is K1 with an initial velocity v1. The final kinetic energy is K2 with a final velocity v2.
We are given that the final velocity (v2) is three times the initial velocity (v1). So, v2 = 3 * v1.
Now, let's find the ratio of the final kinetic energy (K2) to the initial kinetic energy (K1):
K2 / K1 = (1/2) * mass * (v2^2) / ((1/2) * mass * (v1^2))
Since the mass is common on both sides, it cancels out:
K2 / K1 = v2^2 / v1^2
Substituting the value of v2 and v1:
K2 / K1 = (3 * v1)^2 / v1^2
Simplifying:
K2 / K1 = 9 * (v1^2) / v1^2
The v1^2 terms cancel out:
K2 / K1 = 9
Therefore, the kinetic energy increases by a factor of 9 when the velocity increases to three times the original velocity.
To find out by what factor the kinetic energy increases, we need to compare the initial kinetic energy (KE1) with the final kinetic energy (KE2).
The kinetic energy of an object is given by the equation KE = 0.5 * m * v^2, where KE is the kinetic energy, m is the object's mass, and v is its velocity.
Let's assume the initial velocity is v1 and the final velocity is v2. The initial kinetic energy, KE1, can be calculated as KE1 = 0.5 * m * v1^2.
Similarly, the final kinetic energy, KE2, is given by KE2 = 0.5 * m * v2^2.
Given that the final velocity (v2) is three times the initial velocity (v1), we have v2 = 3 * v1.
Substituting this value into the equation for KE2, we get KE2 = 0.5 * m * (3 * v1)^2 = 0.5 * m * 9 * v1^2.
Now we can compare the final kinetic energy (KE2) to the initial kinetic energy (KE1) to find the factor by which the kinetic energy increases:
KE2 / KE1 = (0.5 * m * 9 * v1^2) / (0.5 * m * v1^2)
= (9 * v1^2) / (v1^2)
= 9.
Therefore, the kinetic energy increases by a factor of 9 when the sprinter's velocity increases to three times the original velocity.