A racing dog is initially running at 20.0 m/s, but is slowing down.
Question: How fast is the dog moving when its kinetic energy has been reduced by half?
Question: By what fraction has its kinetic energy been reduced when its speed has been reduced by half?
Use the fact that
kinetic energy = (1/2)MV^2.
For half the kinetic energy, the V^2 term must be reduced from 400 to 200, in this case.
That means V = 14.14 m/s
If you reduce speed by half, you have 1/4 the kinetic energy
To find out how fast the dog is moving when its kinetic energy is reduced by half, we can make use of the formula for kinetic energy:
Kinetic Energy = (1/2) * mass * velocity^2
Let's assume the mass of the racing dog remains constant.
Step 1: Find the initial kinetic energy of the racing dog.
Given:
Initial velocity (vi) = 20.0 m/s
Using the formula above, we can calculate the initial kinetic energy (Ki):
Ki = (1/2) * mass * vi^2
Step 2: Find the final kinetic energy when it is reduced by half.
To find the required velocity, we need to determine the final kinetic energy. Since the kinetic energy is reduced by half, it means the final kinetic energy (Kf) is half the initial kinetic energy (Ki):
Kf = (1/2) * Ki
Step 3: Find the final speed of the dog.
Now that we have the final kinetic energy, we can rearrange the kinetic energy formula to solve for velocity:
Kf = (1/2) * mass * vf^2
Solving for vf:
vf = √[(2 * Kf) / mass]
Step 4: Calculate the final speed.
Substitute the value of Kf into the equation:
vf = √[(2 * (1/2) * Ki) / mass]
Simplify:
vf = √[Ki / mass]
Therefore, the speed of the dog when its kinetic energy is reduced by half is given by vf = √[Ki / mass].
For the second question, the fraction by which the kinetic energy has been reduced when the speed is reduced by half can be calculated as:
Fraction = (Ki - Kf) / Ki
Now, let's solve these step by step:
Step 1:
Given:
vi = 20.0 m/s
Substitute this value into the formula for initial kinetic energy (Ki):
Ki = (1/2) * mass * vi^2
Step 2:
Since the kinetic energy is reduced by half, we can calculate the final kinetic energy (Kf):
Kf = (1/2) * Ki
Step 3:
Rearrange the kinetic energy formula to solve for vf:
vf = √[(2 * Kf) / mass]
Step 4:
Substitute the values of Kf and Ki into the formula for final speed (vf):
vf = √[Ki / mass]
To find the fraction by which the kinetic energy has been reduced when the speed is reduced by half, we can use the formula:
Fraction = (Ki - Kf) / Ki
Substitute the values of Ki and Kf into the formula:
Fraction = (Ki - Kf) / Ki
Simplifying these equations will give you the final answers.
To find the speed of the dog when its kinetic energy has been reduced by half, we need to understand the relationship between kinetic energy and speed. The kinetic energy of an object is directly proportional to the square of its speed. This relationship can be expressed by the equation:
Kinetic Energy = (1/2) * mass * speed^2
Now, let's break down the problem step by step:
Question 1: How fast is the dog moving when its kinetic energy has been reduced by half?
Step 1: Determine the initial kinetic energy of the dog. Since we know its speed, we can calculate the initial kinetic energy using the formula mentioned above.
Step 2: Divide the initial kinetic energy by 2 to find the reduced kinetic energy when it has been halved.
Step 3: Rearrange the formula for kinetic energy and solve for speed. This will give us the revised speed of the dog.
Now, let's move to the next question:
Question 2: By what fraction has its kinetic energy been reduced when its speed has been reduced by half?
Step 1: Determine the initial kinetic energy of the dog using the given speed.
Step 2: Calculate the new kinetic energy using the dog's speed after it has been reduced by half.
Step 3: Divide the new kinetic energy by the initial kinetic energy. This will give us the fraction by which the kinetic energy has been reduced.
By following these steps, we can find the answers to both questions.