Analyzing Energy Changes in Free Fall

Let's analyze what happens to the energy of a body in free fall from a height of 20 meters after it has fallen through a distance of 5 meters.

We'll consider the initial state (at 20 m height) and the state after falling 5 m (at 15 m height).

Initial State: At a height of 20 m

Let the mass of the body be \(m\). Let the acceleration due to gravity be \(g\).

Initial height, \(h_1 = 20\) m.

Initial velocity, \(v_1 = 0\) (since it's a free fall starting from rest).

Initial Potential Energy (PE\( _1 \)): \(\text{PE}_1 = mgh_1 = mg(20) = 20mg\)

Initial Kinetic Energy (KE\( _1 \)): \(\text{KE}_1 = \frac{1}{2}mv_1^2 = \frac{1}{2}m(0)^2 = 0\)

Initial Total Energy (E\( _1 \)): \(E_1 = \text{PE}_1 + \text{KE}_1 = 20mg + 0 = 20mg\)

State After Falling 5 m: At a height of 15 m

The body falls 5 m from the initial height of 20 m. So, its height from the ground is \(h_2 = 20 - 5 = 15\) m.

Potential Energy (PE\( _2 \)) at 15 m height: \(\text{PE}_2 = mgh_2 = mg(15) = 15mg\)

To find the Kinetic Energy (KE\( _2 \)) at this point, we can use the equation of motion under constant acceleration: \(v_2^2 = v_1^2 + 2as\). Here, initial velocity \(v_1 = 0\), acceleration \(a = g\), and displacement \(s = 5\) m.

\(v_2^2 = 0^2 + 2g(5) = 10g\)

Kinetic Energy (KE\( _2 \)): \(\text{KE}_2 = \frac{1}{2}mv_2^2 = \frac{1}{2}m(10g) = 5mg\)

Total Energy (E\( _2 \)): \(E_2 = \text{PE}_2 + \text{KE}_2 = 15mg + 5mg = 20mg\)

Analysis of Energy Changes During Free Fall

We observe that the total energy remains constant: \(E_1 = E_2 = 20mg\). This is consistent with the principle of conservation of mechanical energy in free fall (assuming no air resistance).

Now, let's look at the potential energy change of the body during this free fall.

Initial Potential Energy (PE\( _1 \)): \(20mg\)

Potential Energy (PE\( _2 \)) after falling 5 m: \(15mg\)

Change in Potential Energy: \(\Delta \text{PE} = \text{PE}_2 - \text{PE}_1 = 15mg - 20mg = -5mg\).

The negative sign indicates a loss in potential energy. The amount of potential energy lost is \(5mg\).

How much potential energy was lost compared to the initial potential energy?

Loss in PE \( = 5mg\)

Initial PE \( = 20mg\)

Fraction of initial PE lost \( = \frac{\text{Loss in PE}}{\text{Initial PE}} = \frac{5mg}{20mg} = \frac{5}{20} = \frac{1}{4}\)

So, the body loses one-fourth of its initial potential energy after falling 5 m.

Let's check the options based on our findings about the body's energy during free fall:

• Option 1: Lose one-fourth of its total energy. The total energy remains constant at 20mg due to conservation of energy. It does not lose any total energy (in the ideal case of free fall without air resistance). This option is incorrect.
• Option 2: Lose one-fourth of its potential energy. The initial potential energy is 20mg. The potential energy after falling 5m is 15mg. The loss is \(20mg - 15mg = 5mg\). The fraction of initial potential energy lost is \(\frac{5mg}{20mg} = \frac{1}{4}\). This option accurately describes the change in potential energy. This option is correct.
• Option 3: Gain one-fourth of its potential energy. The potential energy decreases from 20mg to 15mg as the body falls, meaning it loses potential energy, not gains. This option is incorrect.
• Option 4: Gain three-fourth of its total energy. The total energy is conserved and remains 20mg throughout the free fall. It does not gain three-fourth of its total energy. This option is incorrect.

Based on the analysis of energy changes during free fall, the body loses one-fourth of its potential energy after falling 5m from a height of 20m.

Key Concepts: Energy in Free Fall

• In free fall (neglecting air resistance), mechanical energy (sum of potential and kinetic energy) is conserved.
• As the body falls, potential energy decreases because height decreases (\(\text{PE} = mgh\)).
• As the body falls, kinetic energy increases because speed increases (\(\text{KE} = \frac{1}{2}mv^2\)).
• The decrease in potential energy is equal to the increase in kinetic energy, ensuring the total mechanical energy stays constant.

Revision Table: Energy at Different Heights in Free Fall

Additional Information: Conservation of Energy in Physics

The principle of conservation of mechanical energy states that if only conservative forces (like gravitational force or the force of a spring) are doing work on an object or system, the total mechanical energy (the sum of kinetic energy and potential energy) of that system remains constant.

In the context of a body in free fall, gravity is a conservative force. Assuming air resistance is negligible (a non-conservative force), the total mechanical energy remains constant. This means that as the body loses potential energy by decreasing in height, it gains an equal amount of kinetic energy by increasing in speed. The transformation of energy between potential and kinetic forms happens continuously during the free fall.