Q.A body falls towards earth in air. Will its total mechanical energy be conserved during the fall? Justify.

Concept: Conservation of mechanical energy holds only when work is done exclusively by conservative forces.

During the fall, two forces act on the body: gravity (conservative) and air resistance (non-conservative). Gravity does positive work, converting potential energy to kinetic energy. Air resistance opposes motion, doing negative work that dissipates mechanical energy as heat in the surrounding air.

The work–energy theorem gives:

$\Delta E_{\text{mech}} = W_{\text{non-conservative}} = W_{\text{air}} < 0$ …

Why mechanical energy conservation fails here

Mechanical energy—the sum of kinetic and potential energy—remains constant only when the forces doing work are conservative. Gravity is conservative, but air resistance is not. Air resistance is a dissipative force: it opposes motion and converts ordered kinetic energy into disordered thermal energy (heating both the body and the surrounding air). This energy leaves the mechanical "account" entirely, so the total $E_{\text{mech}} = KE + PE$ decreases as the body falls.

Step-by-step justification

1. Identify the forces acting on the falling body.

   Two forces act: gravitational force $mg$ (downward) and air resistance $f_{\text{air}}$ (upward, opposing the velocity).

2. Examine the work done by each force.

   • Gravity does positive work $W_g = mgh$ as the body descends through height $h$, converting gravitational potential energy into kinetic energy.
   • Air resistance does negative work $W_{\text{air}} = -f_{\text{air}} \cdot d$ (where $d$ is the distance fallen), because the force opposes the displacement.

3. Apply the work-energy theorem.

   The net work done equals the change in kinetic energy:

Rearranging:

Since $W_g = -\Delta PE$ (the loss in potential energy), we have:

The left side is the change in total mechanical energy, and it equals the (negative) work done by air resistance.

4. Conclude about energy conservation. …

Concept: Conservation of Mechanical Energy — Only Holds for Conservative Forces

Mechanical energy $E=K+U$ stays constant only when every force doing work

on the body is conservative; a falling body in air also experiences a

non-conservative drag force.

Step 1: List the forces acting on the falling body

Two forces act: gravity $mg$ (downward, conservative) and air

resistance $f_{\text{air}}$ (upward, opposing velocity, non-conservative).

Step 2: Apply the work-energy theorem, separating the two forces

Since $W_g = -\Delta U$ (the drop in potential energy), rearranging gives:

The left-hand side is exactly $\Delta E_{\text{mech}}$, so:

Step 3: Sign of the air-resistance term

Air resistance always opposes the motion, so it does negative work: …