A 2 kg rock is released from rest at a height of 29 m. Ignore air resistance and determine the kinetic energy, gravitational potential energy, and total mechanical energy at each of the following heights;20m 15m 9m
Well, well, well, we have a rock star here! Let's calculate some energies and rock this problem!
At a height of 20 m:
To calculate the kinetic energy, we'll need to know the rock's velocity at that height. Since it was "released from rest," that means it has fallen for some time, gaining speed along the way. But without knowing the time it took to reach that height, we can't say for sure what the velocity is. So, sorry for the anticlimax, but we'll need to skip the kinetic energy for now.
The gravitational potential energy at 20 m can be calculated using the formula:
PE = m * g * h,
where m is the mass of the rock (2 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height (20 m). Plug in the values and you'll get it!
To find the total mechanical energy, we need to add the gravitational potential energy and the kinetic energy (which we've skipped for now). So, for now, the total mechanical energy will be just the gravitational potential energy.
At 15 m and 9 m:
Sadly, my friend, we can't calculate the kinetic energy again without knowing the velocity at those heights. However, we can use the same strategy as before to calculate the gravitational potential energy at each height.
And once again, the total mechanical energy would be the sum of the gravitational potential energy and the kinetic energy (even though we don't know the value of the latter).
Now, isn't this an exciting cliffhanger? But don't worry, with more information, we can rock these calculations and bring the house down!
To determine the kinetic energy (KE), gravitational potential energy (PE), and total mechanical energy (ME) at different heights, we can use the formulas:
KE = 0.5 * mass * velocity^2
PE = mass * acceleration due to gravity * height
ME = KE + PE
Given:
Mass (m) = 2 kg
Initial height (h0) = 29 m
Height at different points (h) = 20 m, 15 m, 9 m
Acceleration due to gravity (g) = 9.8 m/s^2 (approximate)
1. At 20 m:
We can calculate the final velocity (v) using the formula for free-falling objects:
v^2 = u^2 + 2 * g * (h0 - h)
where u = 0 (initial velocity)
Therefore, v^2 = 0 + 2 * 9.8 * (29 - 20)
v = √(2 * 9.8 * 9) ≈ √(176.4) ≈ 13.27 m/s
KE = 0.5 * m * v^2
= 0.5 * 2 * (13.27)^2 ≈ 175.68 J
PE = m * g * h
= 2 * 9.8 * 20 ≈ 392 J
ME = KE + PE
= 175.68 + 392 ≈ 567.68 J
Therefore, at a height of 20 m, the kinetic energy is approximately 175.68 J, the gravitational potential energy is approximately 392 J, and the total mechanical energy is approximately 567.68 J.
2. At 15 m:
Using the same approach as above, we calculate the velocity at this height:
v^2 = 0 + 2 * 9.8 * (29 - 15)
v = √(2 * 9.8 * 14) ≈ √(274.4) ≈ 16.56 m/s
KE = 0.5 * m * v^2
= 0.5 * 2 * (16.56)^2 ≈ 275.02 J
PE = m * g * h
= 2 * 9.8 * 15 ≈ 294 J
ME = KE + PE
= 275.02 + 294 ≈ 569.02 J
Therefore, at a height of 15 m, the kinetic energy is approximately 275.02 J, the gravitational potential energy is approximately 294 J, and the total mechanical energy is approximately 569.02 J.
3. At 9 m:
Following the same steps as above, we calculate the velocity:
v^2 = 0 + 2 * 9.8 * (29 - 9)
v = √(2 * 9.8 * 20) ≈ √(392) ≈ 19.8 m/s
KE = 0.5 * m * v^2
= 0.5 * 2 * (19.8)^2 ≈ 392.04 J
PE = m * g * h
= 2 * 9.8 * 9 ≈ 176.4 J
ME = KE + PE
= 392.04 + 176.4 ≈ 568.44 J
Therefore, at a height of 9 m, the kinetic energy is approximately 392.04 J, the gravitational potential energy is approximately 176.4 J, and the total mechanical energy is approximately 568.44 J.
To determine the kinetic energy, gravitational potential energy, and total mechanical energy at different heights for the 2 kg rock released from rest at a height of 29 m, we can use the formulas for energy.
1. Kinetic Energy (KE): The kinetic energy of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity.
We need to find the velocity at each height, which we can do using the conservation of energy principle. The potential energy at the initial height is equal to the sum of the kinetic energy and potential energy at any given height.
At the initial height of 29 m, the potential energy is equal to the gravitational potential energy.
Gravitational Potential Energy (PE): The gravitational potential energy of an object near the surface of the Earth is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2 on Earth), and h is the height.
Let's calculate the values at each height:
At 20 m:
- The potential energy at this height is given by PE = mgh.
- The gravitational potential energy is PE = 2 kg * 9.8 m/s^2 * 20 m = 392 J.
- We need to find the velocity at this height to calculate the kinetic energy.
- The total mechanical energy (E) is the sum of kinetic and potential energy.
At 15 m:
- The potential energy at this height is given by PE = mgh.
- The gravitational potential energy is PE = 2 kg * 9.8 m/s^2 * 15 m = 294 J.
- We need to find the velocity at this height to calculate the kinetic energy.
- The total mechanical energy (E) is the sum of kinetic and potential energy.
At 9 m:
- The potential energy at this height is given by PE = mgh.
- The gravitational potential energy is PE = 2 kg * 9.8 m/s^2 * 9 m = 176.4 J.
- We need to find the velocity at this height to calculate the kinetic energy.
- The total mechanical energy (E) is the sum of kinetic and potential energy.
To find the velocity at each height, we can use the conservation of energy principle:
Initial potential energy = Potential energy at height + Kinetic energy at height
For example, at 20 m:
Potential energy at 29 m = Potential energy at 20 m + Kinetic energy at 20 m
Using the formulas, we can solve for the velocities:
Potential energy at 29 m = mgh
392 J = 2 kg * 9.8 m/s^2 * 29 m
Solving for h:
h = 392 J / (2 kg * 9.8 m/s^2) = 20 m
So, the velocity at 20 m is 0 m/s because the rock is at rest at this point.
Similarly, you can solve for the velocities and energies at 15 m and 9 m using the same conservation of energy principle.
Once you have the velocities, you can calculate the kinetic energy using the formula KE = (1/2)mv^2 at each height.
Finally, the total mechanical energy (E) at each height is the sum of kinetic and potential energy.
I hope this explanation helps you understand how to calculate the kinetic energy, gravitational potential energy, and total mechanical energy at different heights!