NLM O31

1. Analysis of Terminal Velocity (Falling)

When the ball is falling at its terminal velocity $v_t$, it is in dynamic equilibrium. The net force is zero because the upward air resistance balances the downward gravitational force.

$$ F_{net} = 0 \implies F_{drag} – mg = 0 $$ $$ F_{drag}(v_t) = mg $$

This establishes that at speed $v_t$, the magnitude of the air resistance is exactly equal to the weight $mg$.

2. Analysis Immediately After Bounce (Rising)

The ball bounces elastically. This means there is no loss of kinetic energy, so the speed immediately after the bounce is the same as the speed immediately before.

$$ v_{initial\_up} = v_t $$

However, the direction of velocity is now upwards.

Since air resistance always opposes motion, the drag force now acts downwards. Its magnitude depends only on speed, which is still $v_t$, so the magnitude remains $mg$.

3. Calculating Net Acceleration

Now we sum the forces acting on the ball. Both forces are acting downwards:

• Gravity: $mg$ (Downwards)
• Air Resistance: $F_{drag} = mg$ (Downwards)

Applying Newton’s Second Law:

$$ F_{net} = ma $$ $$ mg (\text{gravity}) + mg (\text{drag}) = ma $$ $$ 2mg = ma $$ $$ a = 2g $$