How fast must a 1,000 kg car be moving to have a kinetic energy of: (a) 2.0 x 10^3, (b) 2.0 x 10^5, (c) 1.0 kW x h?
How high would you have to lift a 1,000 kg car to give it a potential energy of: (a) 2.0 x 10^3, (b) 2.0 x 10^5, (c) 1.0 kW x h?
To calculate the speed of the car for each given kinetic energy, we can use the formula for kinetic energy:
Kinetic energy = 0.5 * mass * velocity^2
(a) For a kinetic energy of 2.0 x 10^3:
2.0 x 10^3 = 0.5 * 1000 kg * velocity^2
Divide both sides of the equation by 0.5 * 1000 kg:
4 = velocity^2
Take the square root of both sides:
velocity = sqrt(4) = 2 m/s
Therefore, the car must be moving at a speed of 2 m/s.
(b) For a kinetic energy of 2.0 x 10^5:
2.0 x 10^5 = 0.5 * 1000 kg * velocity^2
Divide both sides of the equation by 0.5 * 1000 kg:
400 = velocity^2
Take the square root of both sides:
velocity = sqrt(400) = 20 m/s
Therefore, the car must be moving at a speed of 20 m/s.
(c) For a kinetic energy of 1.0 kW x h:
To convert kilowatt-hours to joules, we need to multiply by the conversion factor: 1 kW x h = 3.6 x 10^6 J. So we have:
1.0 kW x h = 3.6 x 10^6 J
3.6 x 10^6 J = 0.5 * 1000 kg * velocity^2
Divide both sides of the equation by 0.5 * 1000 kg:
7200 = velocity^2
Take the square root of both sides:
velocity = sqrt(7200) ≈ 84.85 m/s
Therefore, the car must be moving at a speed of approximately 84.85 m/s.
Now, let's calculate the height required to give the car a potential energy for each given value:
The potential energy formula is:
Potential energy = mass * gravity * height
where gravity is approximately 9.8 m/s².
(a) For a potential energy of 2.0 x 10^3:
2.0 x 10^3 = 1000 kg * 9.8 m/s² * height
Divide both sides of the equation by 1000 kg * 9.8 m/s²:
2.0 x 10^3 / (1000 kg * 9.8 m/s²) = height
height ≈ 0.204 m
Therefore, the car would need to be lifted to a height of approximately 0.204 meters to have a potential energy of 2.0 x 10^3 Joules.
(b) For a potential energy of 2.0 x 10^5:
2.0 x 10^5 = 1000 kg * 9.8 m/s² * height
Divide both sides of the equation by 1000 kg * 9.8 m/s²:
2.0 x 10^5 / (1000 kg * 9.8 m/s²) = height
height ≈ 20.41 m
Therefore, the car would need to be lifted to a height of approximately 20.41 meters to have a potential energy of 2.0 x 10^5 Joules.
(c) For a potential energy of 1.0 kW x h:
To convert kilowatt-hours to joules, we need to multiply by the conversion factor: 1 kW x h = 3.6 x 10^6 J. So we have:
1.0 kW x h = 3.6 x 10^6 J
3.6 x 10^6 J = 1000 kg * 9.8 m/s² * height
Divide both sides of the equation by 1000 kg * 9.8 m/s²:
3.6 x 10^6 / (1000 kg * 9.8 m/s²) = height
height ≈ 367.35 m
Therefore, the car would need to be lifted to a height of approximately 367.35 meters to have a potential energy of 1.0 kW x h.
To determine the speed of a 1,000 kg car in order to have a specific kinetic energy, you can use the formula for kinetic energy:
KE = 0.5 * m * v^2
where KE is the kinetic energy, m is the mass, and v is the velocity or speed of the object.
(a) For a kinetic energy of 2.0 x 10^3 J:
Rearranging the formula, we have 2.0 x 10^3 = 0.5 * 1000 * v^2. Solving for v^2, we get v^2 = (2.0 x 10^3) / (0.5 * 1000) = 4. Therefore, v = √4 = 2 m/s.
(b) For a kinetic energy of 2.0 x 10^5 J:
Using the same formula, we have 2.0 x 10^5 = 0.5 * 1000 * v^2. Solving for v^2, we get v^2 = (2.0 x 10^5) / (0.5 * 1000) = 400. Therefore, v = √400 = 20 m/s.
(c) For a kinetic energy of 1.0 kW x h:
Note that 1 kW x h is equal to 3.6 x 10^6 J (1 kilowatt-hour = 3.6 million joules). Therefore, we can proceed with the kinetic energy calculation using 3.6 x 10^6 J instead of 1.0 kW x h. Using the formula, 3.6 x 10^6 = 0.5 * 1000 * v^2. Solving for v^2, we get v^2 = (3.6 x 10^6) / (0.5 * 1000) = 7200. Therefore, v = √7200 ≈ 84.85 m/s.
For the potential energy of the car, you can calculate it using the formula:
PE = m * g * h
where PE is the potential energy, m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and h is the height.
(a) For a potential energy of 2.0 x 10^3 J:
Rearranging the formula, we have 2.0 x 10^3 = 1000 * 9.8 * h. Solving for h, we get h = (2.0 x 10^3) / (1000 * 9.8) ≈ 0.204 m.
(b) For a potential energy of 2.0 x 10^5 J:
Using the same formula, we have 2.0 x 10^5 = 1000 * 9.8 * h. Solving for h, we get h = (2.0 x 10^5) / (1000 * 9.8) ≈ 20.41 m.
(c) For a potential energy of 1.0 kW x h (which is equal to 3.6 x 10^6 J, as mentioned earlier):
Using the formula, 3.6 x 10^6 = 1000 * 9.8 * h. Solving for h, we get h = (3.6 x 10^6) / (1000 * 9.8) ≈ 367.35 m.
So, to summarize:
(a) Kinetic energy of 2.0 x 10^3 J requires a speed of 2 m/s.
(b) Kinetic energy of 2.0 x 10^5 J requires a speed of 20 m/s.
(c) Kinetic energy of 1.0 kW x h requires a speed of approximately 84.85 m/s.
(a) Potential energy of 2.0 x 10^3 J requires a height of approximately 0.204 m.
(b) Potential energy of 2.0 x 10^5 J requires a height of approximately 20.41 m.
(c) Potential energy of 1.0 kW x h (3.6 x 10^6 J) requires a height of approximately 367.35 m.
A 2-kg block rests on a wall that is 3 m tall. What is the gravitational potential energy of the block? The force of gravity on Earth (g) is equal to 9.8 m/s2.
1/2 m v^2=Ke
v= sqrt 2KE/m
PE=mgh
h= PE/mg