3.3.6 Terminal Velocity

According to Newton's Second Law ($F = ma$), the acceleration of the object is determined by the resultant force: $W - D = ma$.

Initially, $v = 0$, so $D = 0$ and $a = g$. As $v$ increases, $D$ increases, causing the resultant force $(W - D)$ to decrease.

Because the resultant force decreases, the acceleration also decreases over time, even though the object is still speeding up.

When the speed is high enough that $D = W$, the resultant force becomes zero ($0 = ma$), meaning the object continues at a constant velocity.

To calculate terminal velocity, set the acceleration to zero in the equation of motion: $mg - D = 0$.

If the drag force is modeled as $D = kv$ (linear) or $D = kv^2$ (quadratic), substitute this into the equilibrium equation.

For the quadratic model: $mg = kv_t^2$, which leads to the formula for terminal velocity: $v_t = \sqrt{\frac{mg}{k}}$.

To determine $v_t$ experimentally, measure the time taken for an object to pass between two fixed markers in a tall column of fluid once it has stopped accelerating.

Identify the State: If a question mentions 'steady speed', 'constant velocity while falling', or 'maximum speed', immediately apply the condition $F_{net} = 0$.

Graph Interpretation: On a velocity-time graph, the gradient represents acceleration. Terminal velocity is reached where the gradient becomes zero (the line becomes horizontal).

Parachute Scenarios: Remember that opening a parachute increases surface area, which drastically increases drag. This causes a sudden deceleration to a new, lower terminal velocity.

Sanity Check: Always ensure that your calculated terminal velocity is higher for objects with higher mass-to-area ratios (like a lead ball vs. a feather).

Gravity 'Turning Off': A common error is thinking gravity stops acting at terminal velocity. Gravity is constant; it is simply balanced by the upward drag force.

Deceleration vs. Moving Up: When a skydiver opens a parachute, they decelerate rapidly. Students often mistakenly think the skydiver moves upward; in reality, they are still moving down, just at a decreasing speed.

Constant Acceleration: Do not use SUVAT equations ($v = u + at$) for the entire fall, as acceleration is NOT constant when drag is present.