Mechanics Tutorial 7 - Free Fall and Terminal Speed

The acceleration due to gravity on the Earth is 9.81 m/s2.  It is given the physics code g and often the value is approximated to 10 m/s2.  (The latter approximation is perfectly acceptable in A-level examinations.)  All objects will accelerate downwards at this rate, regardless of mass.

Over a short distance, a small light object will hit the ground at the same time as a larger and heavier object.

Over a longer distance, air resistance becomes important.  The air resistance reduces the acceleration.  The faster the object, the greater the air resistance.  Eventually the air resistance balances the weight, and there is zero acceleration.  We have terminal velocity or terminal speed.  Since we are not considering direction, we should call it terminal speed.

For a feather, which has a large surface area compared with its mass, the terminal speed is about 10 cm s-1.  For a sky-diver, it is about 60 m s-1.  For the sky-diver when the parachute opens, it is about 5 m s-1.  The value of the terminal speed depends on not only the mass of the object, but also its surface area.

The concept of terminal velocity can be applied to an object falling in any fluid, i.e. liquid or gas.  It does not apply, of course, to solids.  Nor does it apply in a vacuum.  If you throw an object from an space-craft orbiting the Moon, the object will continue to accelerate at a rate of 1.6 m s-2 until it hits the Moon's surface.  It might be travelling quite fast.

 Question 1

An object falls vertically from a space capsule that is 10 000 m above the Moon's surface. 

(a) Calculate the speed at which the object hits the surface of the moon.  Use g = 1.6 m s-2.

(b) Calculate the time it takes to fall.




Free Fall in Air

Think about a sky divers jumping from a plane:

The animation will help you understand this:

Question 2

 Sketch a speed time graph of a parachutist jumping from an aeroplane, reaching terminal velocity, then opening her parachute.  Explain what is happening at each stage. 



Terminal Speed in Liquids

Dropping objects off tall buildings or lobbing things out of aeroplanes results in certain Health and Safety management issues.  It is rather safer to measure terminal speed in a liquid, since the terminal speed is rather lower.  And it's easier to time as well.

When a ball bearing is dropped into a viscous liquid, it almost immediately reaches its terminal speed, and measuring it is simply a matter of timing the motion between two fixed points a known distance apart.

A viscous material is gooey.  Runny materials have low viscosity.  There are various ways of measuring viscosity, the most common of which is to drop a ball-bearing into the liquid, and measuring its terminal speed.  Remember that at terminal speed, the upwards forces of upthrust and drag and the downwards force of the weight are balanced

The upthrust is the same as the weight of fluid displaced by Archimedes' principle.  Therefore, if the weight is greater than the upthrust, the object will accelerate downwards until the drag balances the difference between the weight and the upthrust.  This is true of all fluids, for example air, or water, or chocolate.

We can calculate the drag force by Stokes' Law, which we will look at in the optional topic, Turning Points in Physics.  The measurement of viscosity is important for the manufacturers of confectionery and lubricating oils.  Measurement of terminal speed is one technique.

Question 3

A small ball bearing is dropped into a cylinder of viscous liquid and allowed to fall until it reaches the bottom.  Describe in as much detail as you can the forces that act on the ball bearing, and explain why it falls at a constant velocity.


Question 4

(a)  Describe how you would measure the terminal speed of a small ball bearing in liquids of different viscosities.

(b) Suggest how the data could be used to determine viscosity.