Question

A 0.21 kg apple falls from a tree to the ground 4.0m below. Ignoring air resistance, determine the apple's kinetic energy, the gravitational potential energy, and the total mechanical energy of the system when the apple’s height above the ground is 3.0m.

Answer 1

Given :

• Mass of the apple (m): 0.21 kg
• Height of the apple above the ground at the start (initial height) (h1): 4.0 m
• Height of the apple above the ground at the end (final height) (h2): 3.0 m
• Acceleration due to gravity (g): 9.81 m/s^2

To Find :

The falling apple's kinetic energy, gravitational potential energy, and total mechanical energy.

Solution :

We can use the conservation of energy principle to find the apple's kinetic energy, gravitational potential energy, and total mechanical energy at the height of 3.0m above the ground.

First, we need to find the gravitational potential energy (GPE) of the apple when it is at the height of 4.0m above the ground:

GPE = mgh

where m is the mass of the apple, g is the acceleration due to gravity (9.81 m/s^2), and h is the height above the ground.

So, GPE = (0.21 kg) x (9.81 m/s^2) x (4.0 m) = 8.2266 J

Next, we find the apple's kinetic energy (KE) just before it hits the ground. We can use the conservation of energy principle, "which states that a system's total mechanical energy is conserved (i.e., it remains constant) if no external forces are acting on it." In this case, gravity is the only force acting on the apple, an internal force within the system (i.e., the apple and the Earth).

So, at the height of 3.0m above the ground, the total mechanical energy (TME) of the system is:

TME = GPE + KE

Since the apple is falling freely, we can assume that all of its potential energy at the height of 4.0m has been converted into kinetic energy just before it hits the ground. Therefore, at the height of 3.0m above the ground, the GPE is:

GPE = (0.21 kg) x (9.81 m/s^2) x (3.0 m) = 6.1359 J

Using the conservation of energy principle, we can find the kinetic energy just before the apple hits the ground:

TME = GPE + KE

KE = TME - GPE = (0.21 kg) x (9.81 m/s^2) x (4.0 m) - 6.1359 J = 2.0907 J

Therefore, the kinetic energy of the apple just before it hits the ground is 2.0907 J.

To summarize:

Gravitational potential energy (GPE) of the apple at a height of 4.0m above the ground: 8.2266 J

Gravitational potential energy (GPE) of the apple at a height of 3.0m above the ground: 6.1359 J

Kinetic energy (KE) of the apple just before it hits the ground: 2.0907 J

The system's total mechanical energy (TME) is conserved throughout the fall.