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3.4 The terminal velocity of a small sphere falling down through a liquid

(balancing forces - objects falling downwards under gravity in fluids)

Apparatus

You can demonstrate the effect of resistive forces in a fluid using the experiment illustrated in the diagram above.

Theory: When an object falls through a fluid there are three forces to take into consideration.

W↓ is the weight of the ball bearing due to gravity (Weight = mass x gravity, W = mg)

U↑ is the upthrust experienced by any object in a fluid. See section 6. Pressure in liquid fluids

F↑ is the resistive force of friction between the ball and the fluid.

W and U are constant and so W-U is a constant and the resultant force in the downward direction.

Initially F is zero at the point where the ball enters the fluid, but, as it descends and speeds up F increases and when F equals W-U the ball bearing descends with uniform velocity - its maximum velocity = terminal velocity.

 

Method:

A large glass tube, sealed at one end with a rubber bung. is filled with a viscous liquid e.g. oil or glycerine (the latter I think is the best and easier to clean out). You need at least a 50 cm depth of liquid.

The glass tube is marked with suitable depth intervals e.g. every 10 cm.

Small steel ball bearings are carefully dropped down a thinner glass tube to make the entry into the fluid as smooth as possible.

The time it takes for the tiny ball to fall between the distance markers is timed.

You need an accurate stopwatch and convert the cm or mm of h into m.

(vertical) velocity (m/s) = d (m) ÷ t (s)

You should find the steel ball falls relatively slowly at first and then attains a maximum velocity - the terminal velocity when the force of the weight of the object is balanced by the resistive forces of friction as the surface of the ball interacts with the liquid.

The 'theoretical results' are shown in the velocity-time graph below.

 

The more viscous the liquid, the smaller the terminal velocity as the friction forces are increasing.

At first when the object starts to fall the accelerating force due to gravity, W↓, is greater than the frictional force slowing it down.

You can tell this from the steep positive gradient at the start where you get the biggest acceleration.

As the speed increases, friction (force F↑, drag effect) increases and this reduces the acceleration.

But, as time goes on, the acceleration decreases (gradient decreasing) because the value of F↑ is increasing.

When all the forces balanced (W-U) = F, the resultant force is zero and the graph becomes horizontal (at time T) and the horizontal value is the maximum speed or terminal velocity.

 

Physics notes index on acceleration of falling objects, experiments, friction, drag effects, gravity, terminal velocity

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Key points force and motion: the terminal velocity of a sphere falling through a liquid

Information sources for Doc Brown's key points: IGCSE-GCSE physics are based on textbooks & syllabus-specifications for students taking the UK AQA, Edexcel, OCR 21st Century Science, OCR Gateway science suite, WJEC, CCEA and CIE GCSE physics 9-1 level science examinations

A structured set of summary revision notes on determining the terminal velocity of a small sphere falling through a liquid, tailored to the major UK GCSE/IGCSE physics exam boards: WJEC, CCEA, CIE, AQA, Edexcel, and OCR. These notes cover theory, experimental method, and exam tips to help students master the topic.

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What Is Terminal Velocity?

• Terminal velocity is the constant speed reached by an object falling through a fluid when the downward force (weight) is balanced by the upward drag force.
• At this point, acceleration stops and the object continues at a steady speed.

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Determining Terminal Velocity in a Liquid

Experimental Setup

• Drop a small sphere (e.g. ball bearing) into a viscous liquid (e.g. glycerol or oil).
• Measure the time taken to fall between two marked points using a stopwatch or light gates.
• Use the equation:
  speed = distance / time e.g. m/s
  to calculate terminal velocity once the sphere reaches constant speed.

Key Variables

Variable Type	Examples
Independent	Type of sphere, mass, radius
Dependent	Terminal velocity
Control	Liquid type, temperature, drop height

Graphical Analysis

• Plot distance versus time or velocity versus time.
• A flat section of the graph indicates terminal velocity.

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Typical Exam Board Specifications

• Focus on forces in fluids, terminal velocity, and motion graphs.
• May include required practicals using spheres in liquids.
• Emphasizes drag forces, streamlining, and balanced forces.
• Includes data analysis and graph interpretation.
• Covers specified practicals on terminal speed using falling objects.
• Links to energy transfer and efficiency.
• Explores Newton’s laws, fluid resistance, and terminal velocity.
• Often includes free-fall scenarios and experimental design.

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Student Tips for Exams

• Define terminal velocity: Constant speed when forces are balanced.
• Describe the experiment clearly: Include apparatus, method, and variables.
• Use graphs: Show how velocity changes and levels off.
• Link to Newton’s laws: Especially the balance of forces.
• Revise practicals: Know how to calculate speed and interpret results.
• Use real-world examples: Ball bearings, skydivers, falling droplets.

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Keywords, phrases and learning objectives for friction between solid an liquid surfaces

Be able to describe and explain an experiment to investigate and determine the terminal velocity of small sphere falling down through liquid - know the method, observations, results analysis and calculations and graph.

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Physics notes index on acceleration of falling objects, experiments, friction, drag effects, gravity, terminal velocity