Lesson 5 of 5Forces

Unit 2 · Lesson 5 of 5

Forces in Action: Air Resistance, Terminal Velocity and Safety

What happens when the air is no longer ignored, why a falling object stops speeding up, and why every piece of safety equipment you have ever used works by making the stop take longer.

7 min read10 practice questionsAsk AI about this →

Key things to remember

1Air resistance opposes motion through the air and grows as the object speeds up; it also grows with frontal area and with a less streamlined shape.
2As a falling object speeds up, drag grows, the net force shrinks and the acceleration shrinks — the object is still speeding up, just more slowly.
3Terminal velocity is reached when drag exactly balances weight: net force zero, acceleration zero, velocity constant and large.
4Opening a parachute multiplies the drag, so the net force points upward while the motion is still downward — rapid slowing to a new, far lower terminal velocity.
5A feather and a hammer fall together only in a vacuum; in air the feather's huge area-to-mass ratio gives it a tiny terminal velocity.
6In a collision the change in velocity is fixed. Extending the stopping time reduces the acceleration, and $F_{n e t} = m a$ then reduces the force by the same factor.
7Crumple zones, airbags, helmets and padded surfaces all work the same way: they make the stop take longer.

Free fall in the kinematics unit assumed the air was not there. That assumption is excellent for a dropped stone over a short distance and hopeless for a parachutist, a leaf or a coffee filter. Putting the air back in needs no new equations — only one more arrow on the free-body diagram, and the willingness to think about what happens to the net force as the object speeds up.

Air resistance is a force that grows with speed

Air resistance, also called drag, is the force the air exerts on an object moving through it. It always points opposite to the object's motion, because it comes from the object having to shove air out of the way. The feature that makes it interesting is that it is not a fixed number: the faster you move, the more air you displace each second, so the harder the air pushes back. A cyclist feels almost nothing at walking pace and a wall of it at $40 km/h$ .

Speed — the dominant factor. Doubling the speed of a car more than doubles the drag on it, which is why highway fuel economy is so much worse than city economy.
Cross-sectional area — the frontal area facing the airflow. A skydiver spread flat has several times the drag of the same skydiver diving head-down.
Shape — a streamlined teardrop lets the air close smoothly behind it; a flat plate leaves a turbulent wake that drags it back.
Air density — thinner air at altitude means less drag, which is why aircraft cruise high and why records are set in mountain velodromes.

Qualitative only in this course

There is a formula for drag, and it is not on your formula sheet and not examined here. What you are expected to do is reason with it: say whether the drag is larger or smaller than before, put its value into a free-body diagram when a question gives you one, and explain what that does to the net force. Every calculation below uses nothing more than adding forces and Newton's second law.

Reaching terminal velocity

Fig 1Velocity against time for a falling object in air. The curve flattens as drag grows to match the weight.

Stage of the fall	Weight	Air resistance	Net force	Acceleration
The instant of release	$m g$ down	Zero — no speed yet	$m g$ down	$9.81 m/s^{2}$ down, the full value
Speeding up	$m g$ down	Growing, still less than $m g$	Downward but shrinking	Downward and decreasing
Terminal velocity	$m g$ down	Exactly $m g$ up	Zero	Zero — the speed stops changing

Read that table as a story. The object leaves your hand with gravity alone acting, so it accelerates at the full $9.81 m/s^{2}$ . As it picks up speed the drag grows, the net force shrinks, and the acceleration shrinks with it — so the object is still speeding up, just less enthusiastically each moment. On a velocity-time graph this shows as a line that starts steep and bends over. Eventually the drag has grown to exactly equal the weight, the net force is zero, and by Newton's first law the velocity stops changing. That final constant speed is the terminal velocity, and the graph flattens into a horizontal line.

Terminal velocity means zero acceleration, not zero velocity

At terminal velocity the skydiver is falling fast — roughly

55 m/s

spread flat, which is about

200 km/h

. What has stopped is not the motion but the change in motion. Nor are the forces zero: there is a large weight down and an equally large drag up. It is the net force that is zero, which is exactly the condition for constant velocity.

Opening a parachute changes one thing only — the area facing the airflow, by an enormous factor. The drag immediately becomes far larger than the weight, so the net force flips to point upward while the skydiver is still moving downward. Opposite signs on velocity and acceleration mean rapid slowing, which is the violent jolt of the opening. As the speed falls the drag falls with it, until the drag once again equals the weight and a new, much lower terminal velocity is reached — around $5 m/s$ , a survivable landing speed. The weight never changed; only the drag did.

The hammer and the feather

On the last Apollo 15 moonwalk in 1971, David Scott dropped a geological hammer and a falcon feather side by side. With no atmosphere to provide drag, they struck the lunar surface together, exactly as Galileo had argued. On Earth the feather loses badly — not because gravity pulls it less, but because its enormous area-to-mass ratio means air resistance balances its tiny weight almost immediately, at a terminal velocity of a few centimetres per second.

Safety technology: the same stop, spread over more time

A collision is a very fast stop. The occupant of a car is travelling at the car's speed, and after the crash they are travelling at zero — that change in velocity is fixed by the physics of the situation and no safety device can alter it. What a safety device can alter is how long the change takes. Because the acceleration is the change in velocity divided by the time it takes, stretching the stopping time shrinks the acceleration, and Newton's second law then shrinks the force in exactly the same proportion.

a = \frac{Δ v}{Δ t} ⟹ F_{n e t} = m a = m \frac{Δ v}{Δ t}

Same mass, same change in velocity. Only the stopping time is free to change — and the force falls in proportion as it grows.

Fig 2The same change in velocity, spread over a longer stopping time, needs a much smaller force.

Technology	What it physically does	Why the force on the body falls
Crumple zone	The front structure folds progressively instead of stopping dead	The cabin takes around $0.15 s$ to stop instead of around $0.02 s$
Airbag	Inflates so the head and chest decelerate against a deep, soft cushion	The occupant stops over tens of centimetres rather than against a rigid steering wheel
Seatbelt with a load limiter	Holds the occupant to the seat, then pays out webbing under very high load	The occupant decelerates with the car, over the car's whole stopping time
Cycling or hockey helmet	A crushable foam liner collapses under impact	The skull stops over the thickness of the liner instead of instantly against the ground
Gym mats and playground surfacing	Compress under a falling body	The fall is arrested over several centimetres instead of on bare concrete
Bending your knees when you land	Your legs fold instead of staying locked	Your torso stops over perhaps $0.5 s$ rather than on impact

A crumple zone is not a weak car

It is a deliberately designed collapsible structure bolted to a rigid safety cell. The ends are meant to be destroyed; the passenger compartment is meant to survive intact. A car built entirely rigid would look better after a crash and would hurt its occupants far more, because the whole stop would happen in the few hundredths of a second it takes the bumper to crush. The visible damage is the safety feature doing its job.

ARigid structure, short stop

Stopping time of a few hundredths of a second
Very large acceleration, since $Δ v$ is fixed
Very large force on the body, from $F_{n e t} = m a$
Little visible damage to the vehicle
Severe injury to the occupants

BCrumple zone, belt and airbag

Stopping time stretched by a factor of five to ten
Acceleration reduced by that same factor
Force on the body reduced by that same factor
The vehicle is visibly wrecked
Occupants survive forces their bodies can tolerate

How to phrase this in an answer

Say it in three linked sentences and the marks are yours. The change in velocity is fixed by the collision. Extending the stopping time reduces the acceleration, because acceleration is change in velocity divided by time. A smaller acceleration means a smaller net force on the occupant, because force equals mass times acceleration. Never just write that the device 'absorbs the shock'.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A skydiver of mass

78 kg

is falling, and at one instant the air resistance acting on her is

340 N

upward. Find the net force and her acceleration at that instant.

Example 2

The same

78 kg

skydiver then reaches terminal velocity. (a) What is the air resistance on her? (b) Her parachute opens and the drag momentarily jumps to

2400 N

. Find her acceleration at that instant.

Example 3

70 kg

driver travelling at

14 m/s

is brought to rest in a collision. Compare the average net force on the driver when the stop takes

0.020 s

with the force when a crumple zone, belt and airbag stretch it to

0.16 s

Where marks get lost

Saying a skydiver at terminal velocity has no forces on her. Two very large forces act — weight and drag — and they cancel. It is the net force that is zero.
Reading terminal velocity as 'stops falling'. The acceleration becomes zero, so the velocity stops changing; she keeps falling at a steady, very high speed.
Claiming heavy objects always fall faster. In a vacuum every object accelerates at $9.81 m/s^{2}$ regardless of mass; in air the differences come from drag and area, not from weight alone.
Explaining a crumple zone by saying it 'absorbs the shock' or 'makes the impact softer'. State the chain: the change in velocity is fixed, a longer stopping time gives a smaller acceleration, and $F_{n e t} = m a$ then gives a smaller force.
Forgetting that when a parachute opens the velocity is still downward while the acceleration points upward. Opposite signs mean slowing down, not moving backwards.

Flashcards 9

Air resistance (drag)

Terminal velocity

What does the velocity-time graph of a falling object in air look like?

Why does a parachute work?

+ 5 more

Practice

A skydiver has reached terminal velocity. Which statement is correct?

80 kg

skydiver falls with

500 N

of air resistance acting upward. What is the magnitude of the net force on her, in newtons?

65 kg

skydiver is at terminal velocity. What is the air resistance acting on her, in newtons?

Immediately after a parachute opens, what are the directions of the skydiver's velocity and acceleration?

60 kg

passenger travelling at

12 m/s

is stopped in

0.15 s

by a seatbelt and airbag. What is the magnitude of the average net force on them, in newtons?

Why does a crumple zone reduce the force on the occupants of a car?

What is the name of the constant speed a falling object reaches when air resistance balances its weight?

A feather and a hammer are released together inside a chamber from which all the air has been pumped out. What happens?

Put the stages of a skydiver's fall in order, from the moment she leaves the aircraft.

1The speed falls until drag equals weight again, at a new much lower terminal velocity
2The canopy opens, drag far exceeds weight, and the net force points upward
3Drag equals weight, so the net force is zero and she falls at a constant terminal velocity
4Drag is zero, so the acceleration is the full $9.81 m/s^{2}$ downward
5Drag grows, so the net force and the acceleration both shrink while the speed still rises

Match each safety feature to what it physically does.

Crumple zone

Airbag

Helmet foam liner

Gym mat

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