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Terminal Velocity

Kinematics

Drop a stone from a great height and it does not keep speeding up forever. After a while it settles into a steady falling speed and stops accelerating altogether. That steady speed is its terminal velocity, and it is reached when the upward push of air resistance exactly balances the downward pull of gravity. It is why skydivers fall at a predictable speed and why a feather drifts down so gently.

How terminal velocity is reached

When an object starts to fall, its weight pulls it down and it accelerates. As its speed grows, so does the air resistance (drag) pushing upward against it. The drag grows roughly with the square of the speed, so it rises quickly. The net force, weight minus drag, shrinks, so the acceleration shrinks too. Eventually the drag equals the weight, the net force becomes zero, and by Newton's first law the object stops accelerating and falls at a constant speed. That speed is the **terminal velocity**. The object keeps falling, but no faster.

What changes terminal velocity

Terminal velocity depends on the object's weight, size, shape and the density of the air. A heavier object needs more drag to balance its weight, so it falls faster before reaching terminal velocity, a bowling ball falls far faster than a beach ball of the same size. A larger surface area or a less aerodynamic shape creates more drag, so it lowers terminal velocity. That is how a parachute works: opening it hugely increases the area and drag, so the skydiver's terminal velocity drops from roughly 53 m/s to about 5 m/s, a speed safe to land at.

Terminal velocity in everyday life

Terminal velocity explains many familiar sights. Raindrops reach terminal velocity after falling only a few metres, which is why rain does not hit the ground at dangerous speeds, a typical raindrop falls at about 9 m/s. A feather reaches its terminal velocity almost immediately and drifts down at under 1 m/s. Skydivers control their terminal velocity by changing shape: tucked head-down they can exceed 90 m/s, while spread-eagle they fall at about 53 m/s. Even hailstones are limited by terminal velocity, though large ones still fall fast enough to cause damage.

Common mistakes

• Thinking terminal velocity means the object stops falling. Correction: it keeps falling, it just stops speeding up and falls at a constant speed.
• Thinking heavier objects always have a higher terminal velocity. Correction: shape and area matter too, a heavy flat object can fall slowly if its drag is large.
• Assuming terminal velocity is reached instantly. Correction: the object accelerates first, and drag builds up gradually until it balances the weight.

Practice questions

1. A falling object has a weight of 50 N and experiences 50 N of air resistance. What is its acceleration?
   Show answer

   0 m/s², the forces balance, so the net force is zero and it falls at terminal velocity.

2. Why does opening a parachute slow a skydiver down?
   Show answer

   It greatly increases the surface area and drag, so the drag exceeds the weight, the net force is upward, and the skydiver decelerates to a new, much lower terminal velocity.

3. A feather and a hammer are dropped on the Moon, where there is no air. Which reaches the ground first?
   Show answer

   They land together, with no air resistance there is no terminal velocity, and both accelerate at the Moon's gravity.

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