A 2.48-kg rock is released from rest at a height of 28.1 m. Ignore air resistance and determine (a) the kinetic energy at 28.1 m, (b) the gravitational potential energy at 28.1 m, (c) the total mechanical energy at 28.1 m, (d) the kinetic energy at 0 m, (e) the gravitational potential energy at 0 m, and (f) the total mechanical energy at 0 m.
PE = m x g x h
PE = 2.48 x 9.8 x 28.1
PE = 682.94J at 28.1m
KE at 28.1m = 0, because the object is still at rest.
The kinetic energy at 0m transfers from the previously stated gravitational potential energy.
KE = 682.94J at 0m
Therefore PE = 0J at 0m
Mechanical energy is the sum of both KE and PE.
At 28.1 meters, the PE = m *g * h
PE=2.48 kg * 9.8 m/s^2 * 28.1 m = 683 J
KE = 0 because the object has not moved.
ME = KE + PE = 683 J
At 0 meters, the KE = PE
KE = 683 J
PE = 0 because PE = m*g*h and h = 0 meters
ME = KE + PE = 683 J
To find the answers to these questions, we need to understand the concepts of kinetic energy, gravitational potential energy, and total mechanical energy. Let's go step by step:
(a) The kinetic energy at 28.1 m:
Kinetic energy is given by the formula: KE = 1/2 * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity of the object.
Since the rock is released from rest, its initial velocity is zero. Therefore, at 28.1 m, the rock will have fallen due to gravity, and its velocity can be calculated using the equation of motion: v^2 = u^2 + 2as, where u is the initial velocity, a is the acceleration (due to gravity), and s is the distance fallen.
In this case, u = 0, a = 9.8 m/s^2 (acceleration due to gravity), and s = 28.1 m (the height the rock has fallen). Plugging these values into the equation, we get:
v^2 = 0 + 2 * 9.8 * 28.1
v^2 = 554.92
v = sqrt(554.92)
v ≈ 23.55 m/s (rounded to two decimal places)
Now, we can calculate the kinetic energy:
KE = 1/2 * m * v^2
KE = 1/2 * 2.48 * (23.55)^2
KE ≈ 659.94 Joules (rounded to two decimal places)
Therefore, the kinetic energy at 28.1 m is approximately 659.94 Joules.
(b) The gravitational potential energy at 28.1 m:
Gravitational potential energy is given by the formula: PE = m * g * h, where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Plugging the given values into the formula, we get:
PE = 2.48 * 9.8 * 28.1
PE ≈ 687.45 Joules (rounded to two decimal places)
Therefore, the gravitational potential energy at 28.1 m is approximately 687.45 Joules.
(c) The total mechanical energy at 28.1 m:
The total mechanical energy is the sum of kinetic energy and gravitational potential energy at a given point.
Total mechanical energy = KE + PE
Total mechanical energy = 659.94 + 687.45
Total mechanical energy ≈ 1347.39 Joules (rounded to two decimal places)
Therefore, the total mechanical energy at 28.1 m is approximately 1347.39 Joules.
(d) The kinetic energy at 0 m:
At 0 m height (ground level), the rock will have fallen the entire height of 28.1 m, and therefore, its velocity will be the same as we calculated earlier (23.55 m/s).
Using the same formula as before:
KE = 1/2 * 2.48 * (23.55)^2
KE ≈ 659.94 Joules (rounded to two decimal places)
Therefore, the kinetic energy at 0 m is approximately 659.94 Joules.
(e) The gravitational potential energy at 0 m:
At 0 m height, the object is at the ground level, so its height (h) is now 0. Therefore, the gravitational potential energy at 0 m is also 0.
(f) The total mechanical energy at 0 m:
The total mechanical energy at 0 m is the sum of the kinetic energy and gravitational potential energy at that point.
Total mechanical energy = KE + PE
Total mechanical energy = 659.94 + 0
Total mechanical energy = 659.94 Joules
Therefore, the total mechanical energy at 0 m is 659.94 Joules.