Terminal Velocity

Fluids

Define terminal velocity.

The constant velocity reached by a falling object when its weight is balanced by the sum of drag and upthrust acting on it.

What forces act on an object falling through a fluid?

Weight acts downwards, while upthrust and drag act upwards.

What happens to a falling object's acceleration when there is a resultant downward force?

It accelerates downwards.

Why does drag increase as a falling object speeds up?

Drag depends on the object's velocity relative to the fluid, so a greater velocity produces a greater drag force.

What happens to the resultant force as a falling object approaches terminal velocity?

It decreases as the drag force increases.

What is the resultant force at terminal velocity?

Zero.

What is the acceleration at terminal velocity?

Zero.

Does an object stop moving when it reaches terminal velocity?

No. It continues moving at a constant velocity.

What force balance occurs at terminal velocity?

Weight = upthrust + drag.

Why is terminal velocity constant?

The forces are balanced, so there is no resultant force and therefore no acceleration.

What is meant by viscous drag?

The frictional force between a solid object and a fluid caused by their relative motion.

What direction does viscous drag act?

Opposite to the direction of the object's motion relative to the fluid.

Define Stokes' law.

For a small sphere moving slowly through a fluid, the viscous drag is F = 6πηrv.

State the equation for Stokes' law.

F = 6πηrv.

In Stokes' law, what does F represent?

The viscous drag force, measured in newtons (N).

In Stokes' law, what does η represent?

The coefficient of viscosity of the fluid, measured in Pa s.

In Stokes' law, what does r represent?

The radius of the sphere, measured in metres (m).

In Stokes' law, what does v represent?

The velocity of the sphere relative to the fluid, measured in m s⁻¹.

According to Stokes' law, how is drag related to velocity?

Drag is directly proportional to velocity.

According to Stokes' law, how is drag related to the radius of the sphere?

Drag is directly proportional to the radius.

According to Stokes' law, how is drag related to viscosity?

Drag is directly proportional to the coefficient of viscosity.

What assumptions are required when using Stokes' law?

The object should be a small sphere moving slowly through a fluid with laminar flow.

What is the equation for the weight of a sphere?

W = ⁴⁄₃πr³ρₛg.

What is the volume of a sphere?

V = ⁴⁄₃πr³.

How is the upthrust on a completely submerged sphere calculated?

Upthrust equals the weight of fluid displaced: U = ⁴⁄₃πr³ρfg.

What does ρₛ represent in terminal velocity equations?

The density of the sphere.

What does ρf represent in terminal velocity equations?

The density of the fluid.

Write the force equation for a sphere falling at terminal velocity.

⁴⁄₃πr³ρₛg = ⁴⁄₃πr³ρfg + 6πηrvterm.

State the equation for the terminal velocity of a sphere falling through a viscous fluid.

vterm = 2r²g(ρₛ − ρf)/(9η).

How is terminal velocity related to the radius of a sphere?

Terminal velocity is proportional to the square of the radius: vterm ∝ r².

What happens to terminal velocity if the radius of a sphere doubles?

It becomes four times greater, assuming all other variables remain constant.

What happens to terminal velocity if the radius triples?

It becomes nine times greater, assuming all other variables remain constant.

How is terminal velocity related to viscosity?

Terminal velocity is inversely proportional to viscosity.

What happens to terminal velocity if viscosity increases?

Terminal velocity decreases.

What happens to terminal velocity if viscosity decreases?

Terminal velocity increases.

How does the density difference between the sphere and fluid affect terminal velocity?

A greater difference between the sphere's density and the fluid's density produces a greater terminal velocity.

Why does a larger sphere generally have a greater terminal velocity?

Terminal velocity is proportional to r², so increasing the radius significantly increases terminal velocity.

At a fixed temperature, does viscosity change when the sphere's velocity changes?

No. The viscosity is a property of the fluid and remains constant at a fixed temperature.

What changes as a sphere speeds up through a fluid: viscosity or viscous drag?

Viscous drag increases, while viscosity remains constant at a fixed temperature.

Why must temperature be controlled in a terminal velocity experiment?

Viscosity depends strongly on temperature.

How does increasing the temperature of a liquid such as glycerine affect its viscosity?

Its viscosity decreases.

How would increasing the temperature of glycerine affect the terminal velocity of a falling sphere?

The terminal velocity increases because the viscosity decreases.

How can terminal velocity be investigated experimentally?

Allow spheres of different radii to fall through a viscous liquid and measure their terminal velocities.

How can the terminal velocity of a sphere be measured using a measuring cylinder?

Measure the time taken for the sphere to travel a known distance between two marks after it has reached terminal velocity, then calculate v = distance/time.

Why should the timing marks be placed below the surface of the liquid?

So the sphere has time to reach terminal velocity before its speed is measured.

What measurements are required in a falling-ball viscosity experiment?

The sphere's radius, the distance between timing marks, the time taken between the marks, and the densities of the sphere and fluid.

How should the radius of a small sphere be measured accurately?

Using a micrometer.

How is terminal velocity calculated from experimental measurements?

vterm = distance travelled/time taken.

What graph can be plotted to investigate terminal velocity using different-sized spheres?

Terminal velocity, vterm, against radius squared, r².

What should a graph of vterm against r² look like if Stokes' law applies?

A straight line through the origin.

Why is vterm plotted against r² rather than r?

Because the terminal velocity equation shows that vterm ∝ r².

What is the gradient of a graph of vterm against r²?

Gradient = 2g(ρₛ − ρf)/(9η).

How can the viscosity of a liquid be determined from a vterm against r² graph?

Use the gradient and rearrange η = 2g(ρₛ − ρf)/(9 × gradient).

Why are the densities of the sphere and liquid needed in the falling-ball experiment?

They are required in the terminal velocity equation to calculate the viscosity.

Why might water be unsuitable for a falling-ball terminal velocity experiment?

Its viscosity is low, so spheres may fall too quickly for accurate differences in terminal velocity to be measured.

Why is glycerine more suitable than water for a falling-ball experiment?

Its higher viscosity makes the spheres fall more slowly, allowing terminal velocities to be measured more accurately.

What happens to the density of most liquids as their temperature increases?

The density decreases because the liquid expands.

What happens to the density of a solid ball bearing as its temperature increases?

It decreases slightly because the solid expands.

How does increasing liquid temperature affect upthrust if the liquid's density decreases?

The upthrust decreases because upthrust depends on the density of the displaced fluid.

Why is it difficult to calculate the terminal velocity of irregular or large objects using Stokes' law?

Stokes' law assumes a small, slowly moving sphere and does not accurately describe complex or turbulent flow around irregular or large objects.

Why can a skydiver's terminal velocity vary?

It depends on factors including body shape, orientation, cross-sectional area and drag.

What happens immediately after a skydiver jumps?

Weight is greater than drag and upthrust, so there is a resultant downward force and the skydiver accelerates.

What happens to the drag on a skydiver as their speed increases?

The drag increases.

What happens when the skydiver's drag and upthrust balance their weight?

The resultant force becomes zero and the skydiver travels at terminal velocity.

Why does opening a parachute reduce a skydiver's terminal velocity?

It greatly increases the area and therefore the drag force, allowing the forces to balance at a lower velocity.

What is the difference between drag and upthrust?

Drag is caused by relative motion through a fluid and opposes that motion, whereas upthrust is caused by displacement of the fluid and acts upwards.