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3.3.7Forces and Motion

Investigating Terminal Velocity

5 min read 9 parts Lesson 18 of 66 1 quiz Virtual lab

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Lesson summary

3.3.7 Investigating Terminal Velocity — OCR A Level Physics A notes: Explain terminal velocity using forces, interpret the velocity–time graph, and plan experiments to measure it.When an object falls through a fluid (air or a liquid), its weight acts downward while drag (and upth…

  • Start (v = 0)
  • Speeding up
  • Worked example 1
  • Worked example 2
  • Worked example 3
  • Resultant force

Section 1 of 9What terminal velocity is

When an object falls through a fluid (air or a liquid), its weight acts downward while drag (and upthrust in a liquid) acts upward.

Drag grows as the speed grows.
Terminal velocity
The object accelerates until the upward forces equal the weight; then the resultant force is zero, the acceleration is zero and the object moves at a constant speed called the terminal velocity vt.

Conditions

  • W = D + U at terminal velocity.
  • In air the upthrust is small and usually ignored, so mg = D.

Section 2 of 9Stages of a fall

1Start (v = 0)
Start (v = 0) no drag; the resultant force is the weight, so a = g.
2Speeding up
Speeding up drag increases, resultant force W − D decreases, so acceleration falls (a = (W − D)/m), but speed still rises.
3Terminal velocity
D = W, a = 0, speed constant.

The velocity–time graph is a curve that starts with gradient g and flattens to a horizontal line at vt.

1
The gradient at any moment is the acceleration.
2
The displacement–time graph curves upward and eventually becomes a straight line whose gradient is vt.
3
If the parachute opens, drag suddenly exceeds weight, the object decelerates, and the speed falls to a smaller terminal value as drag again balances the weight at lower speed.

Section 3 of 9What affects terminal velocity

Drag depends on the speed, the cross-sectional area A, the shape and the density ρ of the fluid.

What affects terminal velocity
For turbulent flow at high speed a good model is D = ½CρAv2 where C is the drag coefficient (this exact formula is not required, but D ∝ v2 is a useful guide).
Setting D = mg gives vt ∝ √(m/A): a heavier object of the same shape has a higher terminal velocity, and a larger area (a parachute) gives a much lower one.
A skydiver face-down has vt near 55 m s−1; head-down with small area, near 80 m s−1.

For slow, smooth flow in a viscous liquid, the drag on a small sphere is proportional to its speed (Stokes' law), D = 6πηrv, with η the viscosity and r the radius, so vt ∝ r2 for a given density.

Section 4 of 9Investigation: ball bearing in liquid

Drop a ball bearing into a tall measuring cylinder of glycerol or washing-up liquid.

Investigation: ball bearing in liquid
Mark equally spaced lines down the tube well below the surface and time the ball between successive marks with a stopwatch or video.
When the times for equal distances become constant, the ball has reached terminal velocity: vt = distance / time.
Repeat for several ball diameters or for different liquids.

Controls

  • same liquid temperature (viscosity depends strongly on it), same liquid, release the ball just under the surface in the centre.
  • Reduce errors by using video and a larger distance, and by timing several balls.
  • The graph of distance against time becomes linear after the transient part.

Another method uses paper cake cases dropped from a fixed height with ultrasonic motion sensors, stacking cases to double the mass and checking that vt2 is proportional to mass if D ∝ v2.

Section 5 of 9Link to energy

At terminal velocity the kinetic energy is constant, so the work done by the weight each second (the power mgvt) is all transferred to thermal energy of the fluid and object by drag. Gravitational potential energy keeps being lost, but kinetic energy does not increase.

Section 6 of 9Interpreting the graphs quantitatively

To find the acceleration at any time from a velocity–time graph, draw a tangent and find its gradient.

1
At the start the gradient equals g; near the end it approaches zero.
2
The terminal velocity is the value at which the curve levels.
3
Because drag depends on speed, the transition is smooth.
4
In experiments with the ball bearing in oil, remember that the oil becomes less viscous when warm, so a faster terminal velocity is measured; keep the temperature steady.
5
Comparing different sized spheres shows how terminal velocity depends on radius, and leads to the idea of a falling-sphere viscometer.

Section 7 of 9Worked examples

  1. A skydiver of mass 80 kg reaches terminal velocity. What is the drag force? What is her acceleration when the drag is 400 N?

  2. A ball bearing falls through 0.40 m of oil at a steady speed in 1.6 s. Find the terminal velocity. If the mass is 0.0040 kg and upthrust is 0.012 N, find the drag.

  3. Drag on a falling object follows D = kv2. Terminal speed is 40 m s−1. What is it if the mass is quadrupled with the same shape?

Section 8 of 9Key terms

TermMeaning
Terminal velocityConstant speed reached when the resultant force on a falling object is zero.
DragResistive force from a fluid, growing with speed.
Resultant forceVector sum of all forces; zero at terminal velocity.
ViscosityA fluid's resistance to flow, important for drag at low speed.
Cross-sectional areaArea facing the oncoming fluid.
StreamliningShaping to reduce drag.

Section 9 of 9Exam technique

1
Saying the object stops accelerating because it stops being pulled by gravity. Gravity still acts; the forces balance.
2
Drawing the velocity–time graph as a straight line then a sudden flat section; it is a smooth curve.
3
Believing a heavier object has the same terminal velocity as a lighter one of the same shape.
4
Forgetting upthrust in a liquid when balancing forces.
5
Saying drag is constant. It increases with speed up to the terminal value.
Try it in the virtual labFree fall: measuring g
Physics 0625 & 9702 SUVAT Mastery Graph Interpretation

Constant Acceleration Equations (SUVAT) & Slopes

v = u + at  |  s = ut + ½at2  |  v2 = u2 + 2as  |  s = ½(u + v)t

  • Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
  • Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
  • Free Fall in vacuum: a = g ≈ 9.81 m/s2 downwards (air resistance negligible).
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