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# Investigate the factors which affect the terminal velocity of a falling object.

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Introduction

AS Physics Coursework                Page  of

Hannan Shah

AS Physics Coursework

HANNAN SHAH

Aim

To investigate the factors which affect the terminal velocity of a falling object.

## Background Knowledge

• Velocity is defined as the rate of change of displacement of an object, with respect to time.  It is the vector quantity corresponding to speed.  Displacement is the distance moved by an object in a particular direction.
• Velocity = Displacement

Time

• An object falling freely under gravity freely under gravity has a constant acceleration, provided the gravitational field strength is constant.  Fluid resistance (liquid or gas, such as air) reduces acceleration.  When fluid resistance equals the object's weight the resultant force is zero as opposing forces are balanced.  When this is the case the velocity becomes and there is no acceleration.  This is the terminal velocity of the object.
• Viscosity is an internal property of a fluid that offers resistance to flow.  An object falling through a viscous medium will reach a terminal velocity when the force of drag acting upwards upon the object equals the force of gravity.
• The terminal velocity of an object depends on the object's size, shape and weight.
• According to Stoke's law, the terminal velocity of a sphere falling through a fluid can be found by the following formula:

4/3πr3g (ρ-σ) = 6πrηvt

OR

vt = 2r2 (ρ-σ) g                η = 2r2 (ρ-σ) g

9η                                  9vt

Where r is radius of the sphere (m), g is acceleration due to gravity (ms-2), η is         viscosity (Ns m-2), σ is fluid density (kg m-3), ρ is sphere density (kg m-3) and vt is         terminal velocity (ms-1).

## Prediction

• I predict that when a ball bearing is dropped in oil it will initially accelerate but this acceleration will reduce gradually until it reaches 0ms-2 and the ball moves with a constant velocity  (it's terminal velocity).
• A ball bearing of a greater radius will fall with a greater velocity and will take longer to reach its terminal velocity.
• A ball bearing of a smaller radius will fall with a smaller velocity and will take less time to reach its terminal velocity.
• The terminal velocity of a ball bearing is inversely proportional to the viscosity of the fluid.

This is a predicted shape for the results graph showing the variation of velocity with respect to time.  This velocity-time graph shows the ball bearing initially accelerating from rest but the acceleration is gradually dropping until it reaches 0ms-2.  This is when the object reaches its terminal velocity.

Hypothesis

• When an object is released from rest the frictional force (F) is 0 and the resultant force is equal to the weight (W=mg) of the object.  When F is less than W the object's velocity increases i.e. there is acceleration.  When an object gains velocity a frictional force opposes the weight of the object and this force grows as velocity increases.  When F=W the resultant force is 0 and there is no acceleration.  The terminal velocity has been reached.
• A ball bearing with a greater radius than another ball bearing should have a greater terminal velocity because, according to Stoke's law, vt is proportional to r2.

vt = 2r2 (ρ-σ) gTherefore, vt∝ r2

9η

• Stoke's law can also be used to explain my prediction that the terminal velocity is inversely proportional to the viscosity of the fluid.  Simply put:

Middle

Measuring cylinderMetre stickStop watchOilRetort stand and clampsStrong magnetThermometerMicrometer

#### Health and Safety

Oil should be handled carefully and any spillages should be cleaned up to keep the environment safe.  A glass thermometer can easily shatter and pose a hazard.  An object like a metre stick can be dangerous if not used correctly.  This will be placed in the middle of the table to ensure it doesn’t pose a hazard.  This will be the precaution taken with all of the equipment.  Most safety hazards are those outside of the experiment such as chairs and other objects.  Such objects will be cleared out of the way to remove any potential risks.

### Method

A micrometer should be used to accurately record the diameter of each ball bearing (in m).  The ball bearings should be weighed (in kg) and their volumes should be worked out using 4/3πr3 (in m3).  This information can be used to work out the density of each ball bearing (in kg m-3) for use with Stoke's law to calculate the terminal velocity of each ball bearing.

Hold a measuring cylinder using a retort stand and clamps.  A metre rule should be used to mark every 10cm on the measuring cylinder with a marker pen.

Conclusion

Percentage Error = Absolute Error  x  100

Value of Quantity

Percentage error in r: 0.005m/0.01m x 100 = 5%

Percentage error in ρ: (5x10-5) kg/0.1kg x 100 = 0.05%

Percentage error in σ: 0.05kg/1kg x 100 = 5%

Percentage error in distance: 0.05cm/50cm x 100 = 0.1%

Time: Absolute error of 0.1 seconds plus reaction times in starting and stopping the         stopwatch, estimated to be 0.5 seconds.

Percentage Error in time = 0.1+0.5s/10s x 100 = 6%

Percentage Error in Vt = 0.1 + 6 = 6.1%

= [Distance] + [time]

Percentage Error in η = 2 x (5+5) + (0.05+5) + (9 x 6.1)

[2r2]         +    [(ρ-σ)]   +        [9Vt]

= 79.95%

A lot of assumptions have been made when calculating the total percentage error, such as the estimated values for each variable when calculating the percentage error, so the large result cannot necessarily be accepted but can be used as a guide when the actual results are recorded.

However, it is clear that a large proportion of the percentage error in η is made up of the percentage error in Vt.  The percentage error in Vt is large due to the large error in time, which is due to the reaction times in starting and stopping the stopwatch.

 BIBLIOGRAPHY Title/Website Author(s) Publisher/Year Physics 1 David SangKeith GibbsRobert Hutchings Cambridge University Press 2000 Heinemann Advanced Sciences - Physics Patrick Fullick Heinemann Educational Publishers 1994 Physics TB AkrilGAG BennetCJ Millar Hodder and Stoughton 1979 Collins Advanced Sciences - Physics Ken DobsonDavid GraceDavid Lovett Collins Educational 1997 www.sciencebyjones.com Larry Jones

HANNAN SHAH

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