Data Collection  

In this experiment the velocity of a ball was measured by analysing a video of its motion. Below is an example of the graph obtained for a slope height of 2.7cm.

In this way the velocity was measured for the ball rolling down the same slope from different heights. The data has been entered into the table below.

The uncertainty in height is estimated from the accuracy of the ruler. 0.2mm was decided upon because it the actual height of the starting position was not very well defined due to the width of the tape used as a marker.

The uncertainty in velocity was estimated by measuring the velocity of the shortest height 4 times.  The values obtained were

0.4273
0.4232
0.4280
0.4244

The uncertainty was then found from (max – min)/2 = 0.0024ms^-1

There was not sufficient time to repeat all measurements.

 

Data Processing 

The uncertainty in v² is calculated from (v_max² - v_min²)/2

 

Data Presentation 

From the best fit line the gradient = 7.243 ms^-2
The gradient of the steepest line = 8.593 ms^-2
The gradient of the least steep line = 6.547 ms^-2

So uncertainty in the gradient  = (8.593 – 6.547)/2 = 1
Gradient = 7 ± 1 ms^-2

 

Conclusion  

The best fit line does pass through all the error bars so we can conclude that v² and h are linearly related.
Theory states that v² = 2gh so the gradient of the v² vs h graph should be 2g so g = gradient/2
g = 3.6 ± 0.5 ms^-2

This is a long way from the accepted value of 9.81ms^-2 so there must be something wrong with our measurements or assumptions.

 

Evaluation  

The error bars are not overly big and the spread of data is not very wide so it does not seem that we have big random errors in out measurements. Repeated measurements of the one height also leads us to the conclusion that random errors were not the cause of our low result for g.

The intercept is quite close to (0,0) when the error bars are taken into account. However there does seem to be some non linear trend to the line.

In deriving the equation v² = 2gh we made the assumption that all the PE will be converted into translational KE, this however is not the case. Some energy will be lost and some will be transformed into rotational KE. This means that the velocity will be less than expected resulting in a lower value of g.   http://hyperphysics.phy-astr.gsu.edu/hbase/rke.html#rke

mgh = ½ mv² + ½ Iω² where ω = v/r
mgh = ½ (m + I/r)v²

So a graph of v² against h will have gradient = 2mg/(m+I/r)
m/(m+I/r) is less than 1 so the gradient will be less than we would have expected had we not taken into account the rotational motion.

Some energy will also be lost so the equation will be

mgh = ½ (m + I/r)v² + E

Sp v² = 2mgh/(m+I/r) – E/(m+I/r)

This implies that the graph of v² – h will have a negative intercept = E/(m+I/r) which supports the results.

On closer inspection of the video analysis we can see that the blue maker dots are not evenly spaced, this seems to imply that the velocity was not uniform however if we look at the graph we see that it this seems to be due to uneven frame speed rather than the motion of the ball.

 

Improvements  

The object of this experiment was to show that v² = 2gh this was not possible due to the rolling motion of the ball. To prevent rolling the ball would need to be slid down a friction free slope. This is not possible but using a hollow steel ball on a slope covered in oil or made of ice might help. It would be interesting to compare the results. The nature of the slope used actually made the rolling effect worse since the point of contact was a circle of smaller radius than the ball (see diag). his acts like the gears on a bike, the number of rotations of the ball as it rolls down the track will be greater than if it rolled down a flat surface. Rolling down a flat plane would therefore reduce the angular velocity but the ball might not roll so straight.

One problem encountered when analysing the video was that in some frames the ball was hard to see. A black background would have made this much better as would a camera with higher resolution ( I used the webcam on my computer). More frames per second would also make the analysis more accurate, I was using 10 frames per second although as mentioned this did not appear to be constant.

On some of the videos the ball was seen to bounce slightly, this could have been reduced by making a smoother transition between the slope and table.

As mentioned it appears that there is actually a non linear trend to the points. To investigate this further the experiment should be repeated with a wider range and smaller steps between the different heights.