What affects the acceleration of a trolley down a ramp?

In Newton's second law of motion he stated force = mass * acceleration. Because these are both equations for force we can say they are equal to each other and write an equation like this.

Ma = mgh/l

In mathematics the first step towards solving an equation is to cancel out components which are the same on both sides. This means getting rid of mass.

A = gh/l

Because gravity and the length are both constants this proves acceleration is directly proportional to height.

This equation proves both my theories; acceleration is directly proportional to height, and mass does not affect acceleration. Of course I have to take into account that all these calculations have been ignoring friction.

I, therefore, can predict that a graph of height over acceleration will be a straight line. This is because acceleration is directly proportional to height and there are no squared terms that would make it a curve.

Another less mathematical way of proving this is to do with gravitational potential energy and kinetic energy.

I say in the background information I have included that gravitational potential energy is the energy something has because of its position. Kinetic energy is the energy a body has because of its movement.

The gravitational potential energy is converted to kinetic energy as it falls. If the object falls from rest as it does in this case then the change in potential energy is equal to the gain in kinetic energy. Therefore

Mgh=mv²/2

As I said before the first step towards solving a mathematical equation is to cancel out components that are the same on both sides.

Gh = v² / 2

This proves that velocity cannot be dependent on mass but because all the other components are constants velocity is directly proportional to height.

Here we have proved that velocity is directly proportional to height, but not acceleration. An equation for acceleration is

Acceleration = change in velocity / time taken.

So acceleration is proportional to velocity, therefore the greater the velocity the greater the acceleration. If velocity is proportional to height, acceleration must also be.

I can say that the graph if height over velocity will be curved. This is because

V² / 2 =10h

Or

V = (20h

This means that there's a squared term involved. When there's a squared term involved the graph is a curve.

You will see the exact graphs I predict in red on the result graphs so I can compare the two in my evaluation.

PRELIMINARY INVESTIGATIONS

So far I know of two factors that should affect acceleration - friction and the height of the slope. Unfortunately friction would prove a difficult variable to do an experiment into because of the minute difference in results from different surfaces and our inadequate timing methods would not be able to find sufficient evidence to support any theory. I have decided to do an experiment into how the height of the slope affects acceleration of a trolley down a ramp. The purpose of this preliminary investigation is to determine the limits of my investigation - the shortest and tallest heights measurable and the most accurate way to time the descent. Also, is it best to measure the descent 3 times, or 5 times?

What's the tallest / shortest height to measure?

I did not consider this worthy of a thorough investigation so I decided to experiment with a ramp and a trolley to see at what height it would move with too much speed and at what height the trolley would move with enough speed. It is important the trolley doesn't move with too much speed because it will be difficult to measure. We do need it too start quite high because we don't want friction to be too high.

I found at 10cm the trolley fell at a slow pace but when I put it up to 20cm the trolley moved at an adequate pace that wouldn't be too hard time. To determine the tallest height I put the trolley at 1/2m. The trolley fell too rapidly and I found it hard to time with a stopwatch. Therefore in my final investigation I shall have the shortest height as 20cm and the tallest height as 45cm. I shall measure every 5cm in between because 5cm make a considerable difference in the speed.

What would be the best method to use to time the descent?

When doing my unofficial investigation into the tallest and shortest heights measurable I noticed that a stopwatch was an unreliable timing method because it relies heavily on my responses I may be getting inaccurate results. I decided to compare the stop clock to light gates. Light gates should be more reliable because they don't rely on human response.

Plan for stopwatch.

Apparatus

Trolley

Ramp

Ruler

Stop clock

Clamps

Metre rule

. Set up the equipment as shown above. The ramp should be held up by the clamps and at a height of 35cm.

2. Measure a distance of 1m from the back of the ramp and mark.

3. Place the trolley at the back of the ramp. Hold in place.

4. Start the stop clock when you let go of the trolley.

5. Stop the stop clock when the back of the trolley passes the 1m line.

6. Repeat 5 times.

Results with stop clock at 35cm.

Height

(cm)

Time 1

(s)

Time 2

(s)

Time 3

(s)

Time 4

(s)

Time 5

(s)

Average

(s)

35

0.75

0.87

0.71

0.90

0.65

0.78

Plan with light gates.

The plan is exactly the same as before but this time set up light gates at the start and finish lines. Attach a piece of cardboard to the top of the trolley so when you let it go it will first break the first light beam and start the timer and then break the finish light beam and stop the timer.

Results with light gates at 35cm

Height

(cm)

Time 1

(s)

Time 2

(s)

Time 3

(s)

Time 4

(s)

Time 5

(s)

Average

(s)

35

0.820

0.800

0.840

0.850

0.820

0.826

Conclusion

As you can see from the results above the results with the light gates seem to be closer together than the results with the stop clock. With the stop clock the results ranged from 0.65 to 0.90. With the light gates only from 0.80 to 0.85. This is a big difference. The light gates seem to be, and should be, more accurate. The light gates also have the advantage of not relying on human responses like the stop clock. This means anyone could set up the light gates but it would have to be the same person every time using the stop clock because of differing response times.

How many times should I do each height?

Looking at my results from the light gates before I don't think it is really necessary for me to do each experiment 5 times because of the accuracy of the light gates and the small range of the results. I will only do each experiment 3 times because I think this is enough to get good results.

PLAN

Apparatus

Trolley

Ramp

Clamps

Metre rule

Light gates

Cardboard

Electronic Scales

Method

. Put the end of the ramp into the clamps this enables you to easily move it up and down to the desired height.

2. Measure a distance of 1 metre along the ramp, the starting line being the length of a trolley away from the end. Mark this on to the ramp.

3. Raise the ramp to the height you are investigating at the end using the clamps. The height should be measured from the start line and on both sides so the ramp isn't lop-sided.

4. Put a light gate in a clamp and put it over the start line so when the trolley goes down the ramp it will break the light beam.

5. Do the same with a light gate for the finish line.

6. Attach a piece of cardboard to the top of the trolley so when it begins it descent it will break the light beam of the first light gate and then continue to break the final light beam of the finish line light gate.

7. Try it out and adjust the light gates if necessary.

8. Find the mass of the trolley and note it down.

9. Place the trolley on the ramp just before the first light gate and let go.

0. The trolley should then descend the ramp, breaking both light beams and giving you a time for its descent. Note this down.

1. Repeat 3 times for each height.

How will I make it a fair test?

I will use the same trolley each time because although the mass won't affect it the differing smoothness of the wheels might speed it up or slow it down because of friction.

I will use the same ramp each time because each ramp will have a different surface smoothness. This will make a difference to the amount of friction and it is important I don't vary this even though it would only make a minuscule difference.

The only other way I can make it as fair as possible is always being very precise in my measurements of height and always doing the experiment in the same conditions e.g. in a classroom all the time rather that sometimes outside in a gale.

RESULTS

Using light gates to time the descent of a trolley down a ramp of a distance of 1m at different heights.

Height

(centimetres)

Time 1.

(seconds)

Time 2.

(seconds)

Time 3.

(seconds)

Average time.

(seconds)

20

.040

.030

.040

.036

25

0.910

0.920

0.970

0.933

30

0.870

0.860

0.870

0.866

35

0.820

0.840

0.800

0.820

40

0.760

0.790

0.810

0.786

45

0.720

0.710

0.730

0.720

CONCLUSION

In this experiment I have predicted and proved that acceleration and velocity of a trolley down a slope is affected by the height of the slope. You can see that this is true by looking at my graphs.

I did various calculation to work out the velocity and acceleration of each height as you will see on the lined paper enclosed.

If you look at my prediction you will see that I predicted that the velocity graph would be a curved graph because of the squared term involved. I also predicted that the acceleration graph would be a straight line because there are no squared terms involved. I can see by looking at my graphs now that I was right. I also went a step further with the acceleration results because, using my ability to resolve components and also my knowledge of gravitational field strength, I could predict exactly what the acceleration would be for each height. I was able to plot my predicted results on a graph and then easily compare them to the results I found.

The graph of my predicted results and the graph of my results aren't that different because the gradient of my predicted results is 1 and the gradient of my results is 0.8. The explanation for this difference is friction, which the predicted results fail to take into account. This means the acceleration must be rising by a steady rate to have a gradient and be a straight line, this is a pattern I have identified. I also notice that in my results of the time it takes before I do any calculations the gap between the results for each height is decreasing. There is no particular amount it decreases by each time because of slightly inaccurate results but is still decreasing. This is reflected in my velocity height graph where between 20 and 25cm heights the difference in velocity is very large and it gradually gets smaller as the height gets bigger bringing the graph round in a curve.

So my results support my original prediction, the higher the slope the higher the acceleration. I know that this is because the only force acting upon a freely falling object starting from a stationary position is its weight, or gravity. Weight differs from mass because weight = mass * gravity. This means that the mass of something is the same anywhere it the universe but the weight isn't because of differing gravity's - the moons gravity is only a 1/6 of the earth's. Gravity/weight acts directly downwards but in this the energy is forced into a different direction. This means only part of g is pulling the trolley downwards; the other is contributing towards friction. To find the force down the slope we need to resolve the components. We do this by multiplying the weight of the trolley by the sine of the angle of the slope - as indicated in the diagram.

Another way of finding SinX is by dividing the height of the slope by the length. In mathematical terms this is Sine = Opposite / Hypotenuse. This makes the formula for the force down the slope mgh/l where h is height and l is length.

In Newton's second law of motion he stated force = mass * acceleration. Because these are both equations for force we can say they are equal to each other and write an equation like this.

Ma = mgh/l

In mathematics the first step towards solving an equation is to cancel out components which are the same on both sides. This means getting rid of mass.

A = gh/l

Because gravity and the length are both constants this proves acceleration is directly proportional to height as you can see in my experiment. This theory is all very well but I have actually proved it in an experiment

EVALUATION

I consider the experiment I did to be fairly accurate and to provide enough evidence to support a firm conclusion. I know that my results are accurate because I could compare them to my predicted results to see whether they were right or not.

There were some anomalous results, however. For example the result for height 45cm on the velocity graph clearly doesn't fit in with the others. The speed is much faster than it should be. This could be because of a decrease in friction. For example, in the other measurements the trolley may have been running along the side thus increasing the friction and lowering the speed but on height 45cm we may have run the trolley down the centre of the slope minimising friction. Another possibility may be an inaccuracy of measurements on our behalf. We may have made the ramp too high and that would increase the speed. Or on the other results the ramp might have been too low decreasing the speed of the trolley. Looking at my acceleration graph it doesn't seem to be height 45cm that is the anomaly but the other results. I think this because it is height 45cm that fits on to the best-fit line and is closer to the predicted result and not heights 25, 35 and 40. There are no really obvious anomalies that I can just rule out of my results as the heights that don't fit onto the line on the velocity graph fit onto the acceleration graph and vice versa.

If we look on the table of results, we can spot some obvious anomalies. The results I consider to be anomalies I have coloured in yellow. On height 25cm one result is 0.970 whereas the other two are 0.910 and 0.920. This result clearly doesn't fit in. it is slower than the other two suggesting an increase in friction or a problem with the light gates. A problem with the light gates is very unlikely because they are a very reliable and precise piece of equipment. My guess would be that there was an increase in friction because it ran along the side of the ramp. The other result I consider being inaccurate is 0.76 at 40cm height. Although this isn't as obvious as the other anomaly it is still quite different in value to the other two results. Any slight difference with the light gates should be noticed, as the light gates are so accurate. It might be faster because there was an inaccuracy in measurement of height or because we pushed it with force down the slope rather than just letting go.

Normally in experiments you find the average of the other results after you have identified and removed the anomalies. You will notice I haven't done this. I didn't do this because I only did each height three times so it could be that the result that seems wrong is right and the other results are wrong. This is why I thought it important that I kept all results. If I were to do this experiment again I would definitely do each height 5 plus times because it would make it easier to identify anomalous results.

I think my results are accurate enough to provide a firm conclusion because I have done everything in my power to make it as accurate as possible. I have used light gates because they were the most reliable timing method available to me and I have always been very careful in my measurements of height. I did the experiment in the same environment each time and used the same trolley and ramp each time. I also have my predicted results to compare my results to so I can make sure they are along the right lines.

If I were to do this experiment again I would be sure to take each height 5 times because of reasons stated above. I would also try and take friction in to account. At the beginning of this experiment I thought because the trolley had such smooth running wheels and the ramp itself had such a smooth surface I wouldn't have to worry about friction. However we can see a big difference between the predicted results without friction and my results. The only factor I can think of that would cause this, aside from air resistance is friction. I could measure friction by putting a trolley on a ramp and then gradually lifting the ramp until the trolley started moving. I could then measure the height of the ramp and use this to work out the amount of friction for each height.

If I had more scientific equipment I would also do an experiment into friction. I could keep the ramp at the same height and use the same trolley and only vary the surface of the ramp. I could use different degrees of sandpaper. I would need very precise timing equipment because there wouldn't be much difference in the times.