Falling Ball Experiment

Diagram

Physics to explain data

An object falling freely under gravity has an acceleration of approximately 9.8 m s ¯². This is known as acceleration due to gravity.

Constant acceleration formulae have been formed for motion in a straight line. For each formula we have four of the following:

* Distance - s

* Time - t

* Acceleration - a

* Initial velocity - u

* Final velocity - v

I will now show how each formula is derived:

Suppose the velocity of the body increase steadily from u to v in time t. The formula would be shown as this:

a = change in velocity

Time taken

Or

a = v - u

t

? v = u +at

Since the velocity is changing steadily, the average velocity is the mean of the initial and final velocities:

a = u + v

2

Average velocity is simply displacement/time. Therefore:

s = u + v

t 2

? S = 1/2 (u + v)t

but:

v = u +at

? s= 1/2 (u + u +at)t

or

s = ut + 1/2 at²

We can rearrange this last formula by substituting the following equation:

t = (v - u)

a

This gives us the final formula:

v² = u² + 2as

For this experiment I have the displacement and the time. I also know the initial velocity because the ball bearing starts from rest and so has an initial velocity of 0 ms. To find the acceleration due to gravity I would use the following formula:

s = ut + 1/2 at²

If the initial velocity = 0 you are left with the following:

s = 1/2 at²

? a = 2s

t²

The acceleration can now be found.

Results

Here are the results I was given:

Height (m)

Time (s)

0.1

0.020

0.2

0.052

0.3

0.061

0.4

0.084

0.5

0.100

0.6

0.110

0.7

0.130

0.8

0.148

0.9

0.192

.0

0.210

.09

0.202

.17

0.224

I will now plot this information into a graph.

Analysis of data

I first plotted a graph of height over time. I found that this gave an exponential graph.

In order to get a straight line graph I plotted height over time². This gave me a straight line through the origin.

I will know use the following formula to explain that the gradient of this graph is exactly half the acceleration:

s = 1/2 at²

? a = 2s

t²

This can be written in the following way:

a = 2( s/t² )

For the graph of height over time², the gradient is found using the following formula:

gradient = s

t²

Therefore the acceleration is twice the gradient of the graph.

Unfortunately the acceleration did not equal exactly 9.8 ms¯². The gradient was 5.105. This means that the acceleration = 10.2 ms¯².

Errors

When measuring the height of the electromagnet errors may have been made. For example the ruler may no have been straight. Also the reading may not be correct due to parallax. The ruler is quite accurate and has an error of ?0.0005 m. Errors when measuring the height can be easily avoided.

The most significant error comes as a result of a delay between the electromagnet being turned off and the metal losing its magnetism. This results in a delay before the ball drops. This will add more time to the results.

I have estimated the acceleration due to gravity to be 9.714ms¯². The actual measurement is 9.81ms¯². I will now calculate the percentage error for the acceleration due to gravity:

9.81 - 9.714 = 0.096

0.096 ? 100 = 9.6

Percentage error = 9.6%

Conclusion

I have found that the height is proportional to time². The gradient of the graph is equal to 1/2 the acceleration. On an ideal graph the gradient would be 4.9