A racing dog is initially running at 20.0 , but is slowing down.
Q1. How fast is the dog moving when its kinetic energy has been reduced by half?
Q2. By what fraction has its kinetic energy been reduced when its speed has been reduced by half?
ke=1/2 m v^2
v=sqrt 2KE/m
so when KE is halved, what is sqrt .5 ?
Q1. To find the speed of the racing dog when its kinetic energy has been reduced by half, we need to know the relationship between kinetic energy and speed.
The formula for kinetic energy is given by: KE = 0.5 * m * v^2
Where:
KE = Kinetic energy
m = Mass of the object
v = Velocity of the object
Since the mass of the racing dog remains constant, we can simplify the equation to:
KE = 0.5 * v^2
If the kinetic energy is reduced by half, we can set up the following equation:
0.5 * v^2 = (0.5 * v_i^2) / 2
Where v_i is the initial velocity of the racing dog.
Solving for v gives:
v = sqrt((v_i^2) / 2)
Substituting the initial velocity v_i = 20.0 m/s:
v = sqrt((20.0^2) / 2)
v ≈ 14.14 m/s
Therefore, when the kinetic energy is reduced by half, the dog is moving at a speed of approximately 14.14 m/s.
Q2. To find the fraction by which the kinetic energy has been reduced when the speed is reduced by half, we can use the relationship between kinetic energy and velocity.
Let's assume the initial kinetic energy is KE_i and the final kinetic energy is KE_f.
We know that kinetic energy is proportional to the square of velocity. Therefore, we can write:
KE_i = 0.5 * m * v_i^2
KE_f = 0.5 * m * v_f^2
Since we are given that the speed (velocity) has been reduced by half, we can write:
v_f = v_i / 2
Substituting the value of v_f into the equation for KE_f:
KE_f = 0.5 * m * (v_i / 2)^2
= 0.5 * m * (v_i^2 / 4)
= 0.125 * m * v_i^2
To find the fraction by which the kinetic energy has been reduced, we can calculate:
KE_f / KE_i = (0.125 * m * v_i^2) / (0.5 * m * v_i^2)
= (0.125 * v_i^2) / (0.5 * v_i^2)
= 0.125 / 0.5
= 0.25
Therefore, when the speed is reduced by half, the kinetic energy is reduced by a fraction of 0.25 or 1/4.
To answer both questions, we need to understand the relationship between kinetic energy, speed, and mass.
1. How fast is the dog moving when its kinetic energy has been reduced by half?
The formula for kinetic energy is given by: KE = (1/2) * m * v^2, where KE represents the kinetic energy, m represents mass, and v represents the velocity (speed) of the object.
In this case, the mass of the dog is not given, but it's not required since we are only comparing the change in kinetic energy. So we can compare the velocities directly.
Given that the initial velocity is 20.0 m/s, let us assume the final velocity when the kinetic energy is reduced by half is v_final.
We can set up the equation for kinetic energy:
(1/2) * m * (20.0^2) = (1/2) * m * (v_final^2)
Now, let's solve for v_final:
20.0^2 = v_final^2
400 = v_final^2
Taking the square root of both sides:
v_final = 20.0 or -20.0
Since the speed cannot be negative, the dog's speed would be 20.0 m/s when its kinetic energy has been reduced by half.
2. By what fraction has its kinetic energy been reduced when its speed has been reduced by half?
For this question, we need to calculate the ratio of the final kinetic energy (KE_final) to the initial kinetic energy (KE_initial) when the speed is reduced by half.
The formula for kinetic energy is the same as mentioned before - KE = (1/2) * m * v^2.
Let us assume the initial kinetic energy is KE_initial, and the final kinetic energy is KE_final.
When the speed is reduced by half, the final speed (v_final) would be (20.0 / 2) = 10.0 m/s.
Now, let's calculate KE_initial and KE_final:
KE_initial = (1/2) * m * (20.0^2)
KE_final = (1/2) * m * (10.0^2)
To calculate the fraction by which the kinetic energy has been reduced, divide the final kinetic energy by the initial kinetic energy:
Fraction = KE_final / KE_initial
Fraction = [(1/2) * m * (10.0^2)] / [(1/2) * m * (20.0^2)]
The mass (m) cancels out, reducing the equation to:
Fraction = (10.0^2) / (20.0^2)
Fraction = (10.0^2) / (400)
Fraction = 100 / 400
Simplifying the fraction:
Fraction = 1 / 4
Therefore, the kinetic energy has been reduced by 1/4 (or one-fourth) when the speed is reduced by half.