A racing dog is initially running at 15.0 m/s, but is slowing down.How fast is the dog moving when its kinetic energy has been reduced by half? By what fraction has its kinetic energy been reduced when its speed has been reduced by half?
A car travels at a constant speed around a circular track whose radius is 2.8 km. The car goes once around the track in 300 s. What is the magnitude of the centripetal acceleration of the car?
Reducing V by a factor 1/sqrt2 = 0.70711 reduces kinetic energy by half.
If you reduce speed by half, KE decreases to 1/4 the former value.
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To answer the first question, we need to find the new speed of the dog when its kinetic energy has been reduced by half.
Let's start by understanding the equation for kinetic energy:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Given that the initial velocity (v1) is 15.0 m/s and the new kinetic energy (KE2) is half of the initial kinetic energy (KE1), we can set up the following equation:
(1/2) * mass * v1^2 = (1/2) * mass * v2^2
Since the mass is constant and cancels out, we can simplify the equation to:
v1^2 = v2^2
Taking the square root of both sides of the equation, we get:
v1 = v2
Therefore, the speed of the dog when its kinetic energy has been reduced by half is equal to its initial speed - 15.0 m/s.
Now, let's tackle the second question - finding the fraction by which the kinetic energy has been reduced when the speed is reduced by half.
We know that kinetic energy is directly proportional to the square of the velocity. So, when the velocity is reduced by half (v1/2), the new kinetic energy would be:
KE2 = (1/2) * mass * (v1/2)^2
Simplifying the equation, we get:
KE2 = (1/2) * mass * v1^2 * (1/4)
Dividing the new kinetic energy (KE2) by the initial kinetic energy (KE1), we find:
KE2/KE1 = [(1/2) * mass * v1^2 * (1/4)] / [(1/2) * mass * v1^2]
Simplifying further, we get:
KE2/KE1 = (1/4) / 1
KE2/KE1 = 1/4
Therefore, the kinetic energy has been reduced by one-fourth (1/4) or 25% when the speed of the dog is reduced by half.