K323 / 2027
Dynamics overview

Lesson 7 of 7

Falling with air resistance

  • I can explain motion in a uniform gravitational field with and without air resistance, including terminal velocity.Syllabus

    Syllabus K323, 3(m). Explain motion in a uniform gravitational field with and without air resistance, including terminal velocity.

A falling object can speed up while its acceleration gets smaller. Increasing air resistance reduces the downward resultant force.

Weight is mg, and acceleration comes from resultant force / mass. Take downwards as positive in the examples on this page.

Without significant air resistance

For a body of constant mass in a uniform gravitational field, weight is constant. If air resistance is negligible, weight is the only force, so:

a = Fresultant / m = mg / m = gNear Earth, this is approximately 10 m/s2 downwards. A released body gains about 10 m/s of downward velocity each second while this model applies.

The mass cancels, which explains why objects of different masses have the same free-fall acceleration in this model. Gravity has not disappeared: it is the force causing the acceleration. With no air resistance, this model does not produce a terminal speed.

As a body falls through still air

Consider the same body released from rest, with its shape and orientation unchanged. At release its speed relative to the air is zero, so air resistance is zero. As its downward speed increases, upward air resistance grows, while its weight remains the same.

Same object and same weight throughout. Each snapshot uses the same force scale.

1. Released from rest

Selected body: 0.50 kg falling object

1. Released from restThe selected body is the same 0.50 kg object. Its weight is 5.0 N downwards. Air resistance is 0.0 N upwards. There is no upward force arrow because air resistance is zero at release in still air. The resultant is 5.0 N downwards and acceleration is 10.0 metres per second squared downwards.5.0 NWeight by Earth

Air resistance: 0 N. Resultant: 5.0 N downwards.

2. A later instant

Selected body: 0.50 kg falling object

2. A later instantThe selected body is the same 0.50 kg object. Its weight is 5.0 N downwards. Air resistance is 2.0 N upwards. The upward arrow uses the same force scale as the weight. The resultant is 3.0 N downwards and acceleration is 6.0 metres per second squared downwards.5.0 NWeight by Earth2.0 NDrag by air

Resultant: 3.0 N downwards. The object is still speeding up.

3. Terminal motion

Selected body: 0.50 kg falling object

3. Terminal motionThe selected body is the same 0.50 kg object. Its weight is 5.0 N downwards. Air resistance is 5.0 N upwards. The upward arrow uses the same force scale as the weight. The resultant is 0.0 N and acceleration is zero. The object continues downwards at a non-zero constant velocity.5.0 NWeight by Earth5.0 NDrag by air

Equal forces. Zero acceleration; non-zero downward velocity.

A constant-mass body at three stages: release, a later downward-moving instant, and terminal motion. Weight stays at 5.0 N; the supplied upward air resistances are 0 N, 2.0 N and 5.0 N.

Put these in order

A skydiver jumps from a hovering helicopter. Put the explanation of how the skydiver reaches terminal velocity in order.

Show the answer

1. At the start the skydiver is at rest, so air resistance is zero and the resultant force equals the weight. A resultant force equal to the full weight gives the largest acceleration, g.

2. The skydiver accelerates downwards at g and gains speed. Air resistance grows with speed, so moving faster brings a larger upward force.

3. As the speed rises, air resistance increases. The weight is unchanged, so a larger upward force leaves a smaller downward resultant.

4. The resultant force, weight minus air resistance, gets smaller, so the acceleration decreases. The speed still rises, but more slowly, until the two forces are equal.

5. Air resistance becomes equal to the weight, so the resultant force is zero. Zero resultant force means zero acceleration, so the velocity stops changing.

6. The skydiver falls at a constant velocity: terminal velocity.

Worked example

Follow the resultant, not just the weight

The object has mass 0.50 kg and g = 10 N/kg. Its weight is 5.0 N throughout. Subtract the upward air resistance from the downward weight.

Downwards is positive; mass is 0.50 kg
Air resistance / NDownward resultant / NAcceleration / (m/s2)
05.0 - 0 = 5.05.0 / 0.50 = 10
2.05.0 - 2.0 = 3.03.0 / 0.50 = 6.0
5.05.0 - 5.0 = 00 / 0.50 = 0

At the middle stage, acceleration is still downwards, so the downward speed is still increasing. It increases less rapidly than at release because the resultant has fallen from 5.0 N to 3.0 N.

Try it step by step

Work it out in steps: Weight in N, then Resultant force (weight minus air resistance) in N, then Acceleration in m/s2.

  1. Problem 1. A 0.60 kg ball is falling while air resistance of 2.0 N acts upwards on it. Take g = 10 N/kg. Find its acceleration at this instant.
  2. Problem 2. A 2.0 kg parcel is falling while air resistance of 5.0 N acts upwards on it. Take g = 10 N/kg. Find its acceleration at this instant.
  3. Problem 3. A 0.80 kg shuttlecock is falling while air resistance of 6.0 N acts upwards on it. Take g = 10 N/kg. Find its acceleration at this instant.
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Problem 1.

Weight in N: W = 0.60 x 10 = 6.0 N

Resultant force (weight minus air resistance) in N: F = 6.0 - 2.0 = 4.0 N downwards

Acceleration in m/s2: a = 4.0 / 0.60 = 6.7 m/s2 (2 sf)

Problem 2.

Weight in N: W = 2.0 x 10 = 20 N

Resultant force (weight minus air resistance) in N: F = 20 - 5.0 = 15 N downwards

Acceleration in m/s2: a = 15 / 2.0 = 7.5 m/s2 (2 sf)

Problem 3.

Weight in N: W = 0.80 x 10 = 8.0 N

Resultant force (weight minus air resistance) in N: F = 8.0 - 6.0 = 2.0 N downwards

Acceleration in m/s2: a = 2.0 / 0.80 = 2.5 m/s2 (2 sf)

Terminal velocity

When air resistance equals weight, the resultant and acceleration are zero. The object continues downwards at a constant, non-zero velocity called its terminal velocity, while these conditions remain unchanged.

Velocity increases while acceleration decreases

Downwards is positive. This is a schematic graph.

A falling object approaches terminal velocityA schematic velocity-time graph takes downwards as positive. Starting at zero velocity, the curve rises steeply and then becomes progressively flatter, approaching a positive, non-zero terminal velocity. Its gradient, the downward acceleration, decreases towards zero. No numerical times or terminal speed are specified by this graph.Downward velocityTime0Terminal velocity(non-zero)

The curve approaches a level above zero. The force snapshots do not supply numerical times or speeds.

For release from rest, the downward velocity rises while the curve becomes less steep. Its gradient approaches zero as the velocity approaches a constant terminal value. This is a schematic graph: no numerical times or speeds are specified.

Read the two features separately: increasing graph height means increasing downward velocity; decreasing gradient means decreasing acceleration. A horizontal line at a positive velocity represents steady downward motion, not rest.

The supplied force values let us calculate acceleration at the three stages. They do not tell us the times of those stages or the numerical terminal velocity. A real falling object may reach the ground before it approaches terminal motion.

What if the air resistance changes?

If an already downward-moving object suddenly experiences air resistance greater than its weight, the resultant is upwards. Its acceleration is then upwards, so it slows while still travelling downwards. Zero acceleration occurs only when the forces balance.

Terminal velocity does not mean zero weight or zero velocity. Both weight and air resistance still act. Their balance gives zero acceleration and a constant downward velocity.

Check your understanding

An object is falling downwards at terminal velocity through still air. Which explanation is correct?

  1. Air resistance balances weight, so acceleration is zero while it keeps moving.
  2. Air resistance has removed the object's weight, so there is no force.
  3. Air resistance is now greater than weight, so the speed stays constant as it falls.
Show the answer

Air resistance balances weight, so acceleration is zero while it keeps moving.

Zero resultant means zero acceleration. The existing downward velocity is non-zero and remains constant while these conditions persist.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

A skydiver jumps from a hovering helicopter. Explain why her speed increases at first but later stays constant. [3]

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