The Relationship between the Angle of Elevation of a Ramp and the Speed of a Ball


Introduction

In this piece of coursework I'm going to investigate and measure the
speed of the ball rolling down a ramp. From the data that I'm going to
collect I'm going to be able to work out the Gravitational potential
energy when changing the height, the friction force acting on the ball
whilst it rolling down, and finally the kinetic energy exerted by the
ball.

Planning

Fair Testing

Before I would start the experiment I would have to devise a suitable
method that would provide a safe test, which is fair and accurate. In
this experiment I'm going to have to roll down a ball whilst changing
the angle of elevation of the ramp, and the factors that would have an
effect on the accuracy of the output results are;

Ø Size of the ball

Ø Heaviness of the ball

Ø Length of the ramp

Ø Height of the ramp

Ø Texture of the ball

Ø Smoothness of the ramp

Ø Angle of ramp from the ground surface

Here in this experiment the only factor that I'm going to change would
be the height of the ramp which in turn would affect the angle of the
ramp too, if the length of the ramp stays constant.

For the experiment to be a fair one I have to;

Ø Keep all the factors that affect the experiment constant except the
ones that I'm testing.

Ø All the experiments have to be done by the same person on the same
day, because if we changed the person that is doing the experiment it
may lead to different reaction times making the experiment inaccurate

Ø All I have to change is the height of the of the ramp from the
ground it's resting on. And by keeping the length of the ramp constant
that would in turn proportionally change the angle of the ramp.

If I managed to keep all the factors that I need constant, constant
and would provide expected results then my experiment would be a fair
and accurate test.


Preliminary Work

Golf Ball

Height of Slope

Angle of slope

Time Taken (s)

1

2

3

4

6

1.7

4.1

4.24

4.19

4.16

9

2.58

3.38

3.26

3.25

3.31

12

3.44

3.1

2.92

2.19

3.03

15

4.3

2.69

2.72

2.69

2.68

18

5.16

2.44

2.36

2.33

2.37

Squash Ball

Height of Slope

Angle of slope

Time Taken(s)

1

2

3

4

6

1.7

4.87

4.79

4.79

4.73

9

2.58

3.74

3.75

3.76

3.74

12

3.44

3.34

3.25

3.27

3.24

15

4.3

2.89

2.83

2.89

2.91

18

5.16

2.60

2.67

2.53

2.62

Table Tennis Ball

Height of Slope

Angle of slope

Time Taken(s)

1

2

3

4

6

1.7

Went

4.53

5.08

5.26

9

2.58

Off

3.88

/

3.91

12

3.44

course

15

4.3

18

5.16

Tennis Ball

Height of Slope

Angle of slope

Time Taken(s)

1

2

3

4

6

1.70

6.95

5.67

6.05

5.79

9

2.58

4.65

4.91

4.79

4.82

12

3.44

3.75

3.72

3.69

3.67

15

4.30

3.22

3.19

3.12

3.18

18

5.16

2.75

2.89

2.75

2.79

Above are the preliminary results.

Selecting the most Suitable Apparatus

When looking at the preliminary results the golf ball traveled too
fast for us to be able to stop the time accurately, the table tennis
ball was too light so has gone off course most of the time, due to the
air resistance, and that was enough to deflect it. The tennis ball was
too big and I don't think it would be a good idea to use it, as we
would find it hard to place correctly on the starting point line, and
it would take too much time which might lead to anomalies. But the
squash ball to me was the most suitable ball to use, and it stays in
its path all time which is in a straight line down the ramp. In the
preliminary tests we used 3 cm think books to lift up the ramp, but I
don't think that that would be a good idea, because the books might be
compressed a bit, which would decrease the height we are aiming to
achieve. So I think I'm going to use building blocks that are 8.2cm in
height each and that would cancel out the chance of any change in the
height due to compression. When thinking about time, the most accurate
stopwatch we can get would be a 2 d.p. stopwatch and that's all we've
got. There are lots of different types of ramps out there, with
different types of surfaces but the only one that was available to me
was a wooden ramp that has a groove going through it, to provide a
path for the ball to roll through. Below is the apparatus that I'm
going to use to do the experiment.

Apparatus

* Ramp that has been graved in a straight line to provide a course
for the ball to roll through. (length of ramp is 200cm)

* Squash Ball (24.43 grams)

* Stop Watch (2.d.p accuracy)

* Building Blocks that are 8.2 cm in height each.

Prediction

I predict that the increase in the ramps angle is proportional to the
speed of the ball. So if the angle increases the speed of the ball
increases too. If the ramp was completely horizontal (angle 0o) the
velocity of the ball would be zero as there would be no way of the
gravitation pulling it through downwards, but if the ramp was put to a
small gradient the ball would roll down slowly, and as you increase
the gradient the speed of the ball increases too. But when the ramp is
put vertically (900) the ball would free fall at the speed of gravity
(9.81 m/s), due to the ramp not being there to put friction against
the ball, and to deflect its course to another direction. As I would
get different speeds, as I differentiate the steepness of the ramp, I
will have to work out the mean speed using an equation which is the
following;

Mean speed (m/s) = Distance Traveled (m) X Mean Time Taken (S)

Secondly I predict that as the Gravitational potential energy
increases, the kinetic energy also increases proportionally

Thirdly I predict that as the height of the ramp increases the G.P.E
also increases proportionally

Fourthly I predict that as the speed of the ball (squared) increases
the kinetic energy exerted is also increased proportionally.

Fifthly I predict that as the kinetic energy increases the friction
force also increases, and that the friction force is greater than the
kinetic energy.


Number

Predication

1

As the angle of the ramp increases the speed of the ball increases
proportionally.

2

As the Gravitational potential energy increases, the kinetic energy
also increases proportionally

3

As the height of the ramp increases the G.P.E also increases
proportionally

4

As the speed of the ball (squared) increases the kinetic energy
exerted is also increased proportionally.

5

As the kinetic energy increases the friction force also increases, and
that the friction force is greater than the kinetic energy.

So to work out the G.P.E I would use this formula;

G.P.E (j) = Weight (N) X Change in Height (m)

But to work out the weight I would first have to use this formula;

Weight= Mass (kg) X Gravity (N/Kg) Where the gravity is constant at
9.81N/Kg

To work out Kinetic energy I would use this formula;

Ek (j) = ½ Mass (Kg) X Speed2 (m/s2)

And finally to work out the Friction force I would use;

Friction Force (j) = G.P.E (j) - K.E. (j)

Number & Range of Readings

As my main non constant value would be the height, I will have to
choose the heights that I'm able to have using the building blocks. I
will have six different heights which I think would give me a large
range of results. The heights would be 8.2, 16.4, 24.6, 32.8, 41 and
49.2 centimeters. To work out how big the angles are I have to use
this formula;

Angle = sin-1(Opp/Hyp)

Angle = sin-1(height of ramp/200)

So the angles of the heights are;

Height (cm)

8.2cm

16.4cm

24.6cm

32.8cm

41cm

49.2cm

Angle (Ø0)

2.3490

4.7030

7.0650

9.4390

11.8290

14.2410

For each height I think a suitable number of readings would be five of
each height, which would make any anomalous results appear clearly,
and would also when taken out average would get rid or minor errors


Methodology

Firstly I would collect the apparatus, and then I would set it up. (As
shown below.) then I would with a red marker, mark down the starting
point, then the end point which is exactly 200cm. I did this so that
I'd be able to start the ball from the same position, and end it the
same for all the tests. After it being set up, I draw an empty table
of results, so that I would fill it in as I do each experiment. Then I
would place the squash ball which is 24.43g and has a diameter of
3.9cm. at the starting point, and hold it with one hand, and have my
other hand on the stop watch so that I let go of the ball, and start
the stop watch together, to get rid of ad much human error, then I
would stop the stopwatch as the ball crosses the finish line, which
would be placed 200cm after the starting line. I would repeat that for
every height respectively, five times each. After I finish all the
heights, and all their repeats, I would take apart the apparatus
carefully, and place each part in its designated place.

[IMAGE]

Safety

One of the most important aspects of the experiment is safety. As it
would be the safety of my colleagues in the class. So I would have to
handle every thing very cautiously incase the ramp or the building
blocks fall on someone and injure them. So I will have to do my best
to avoid such accidents from happening.

Obtaining Your Evidence

Results;

Height

Angle

Time Taken

Speed average

(cm)

1

2

3

4

5

Average

m/s

8.2

2.349

4.37

4.44

4.44

4.41

4.41

4.414

0.453

16.4

4.703

2.75

2.8

2.81

2.84

2.88

2.816

0.710

24.6

7.065

2.37

2.37

2.44

2.44

2.39

2.402

0.832

32.8

9.439

1.96

1.97

2.03

2.03

2.03

2.004

0.998

41

11.829

1.78

1.78

1.78

1.85

1.82

1.802

1.109

49.2

14.241

1.75

1.75

1.71

1.72

1.69

1.724

1.160

Weight = 0.2343 X 9.81

Weight = 2.298483 Newton's

G.P.E Results; G.P.E (j) = Weight (N) X Change in Height (m)

Height

Weight

G.P.E

(m)

(N)

(J)

0.082

2.29848

0.188476

0.164

2.298483

0.376951

0.246

2.298483

0.565427

0.328

2.298483

0.753902

0.41

2.298483

0.942378

0.492

2.298483

1.130854

K.E Results; Ek (j) = ½ Mass (Kg) X Speed2 (m/s2)

Mass

Speed^2

K.E.

(Kg)

(m/s)^2

(J)

0.2343

0.205303

0.024051

0.2343

0.504423

0.059093

0.2343

0.693288

0.081219

0.2343

0.996012

0.116683

0.2343

1.231829

0.144309

0.2343

1.345815

0.157662

Friction Force Results; Friction Force (j) = G.P.E (j) - K.E. (j)

G.P.E

K.E

Friction Force

(J)

(J)

(J)

0.188476

0.024051

0.164424357

0.376951

0.059093

0.317858083

0.565427

0.081219

0.484208072

0.753902

0.116683

0.637219622

0.942378

0.144309

0.798069265

1.130854

0.157662

0.973191371


Analyzing your Evidence & Drawing Conclusions

Graph Number

Graph

1

Angle of Ramp against Speed of Ball

2

G.P.E against K.E

3

Height of ramp against G.P.E

4

Speed of Ball squared against Kinetic energy exerted

5

Kinetic energy against Friction Force

Graph number one, provides evidence for my first prediction, which is
that as the angle of elevation increase the speed of the ball
increases too. And the graph shows that by starting first starting
off, and going in a positively skewed straight line. This happened
because as the angle increases the G.P.E increases and the friction
force acting on the ball decreases because the gravity force
strengthens.

Graph number two, shows that as the G.P.E increase the Kinetic energy
increases. Because if we increase the height of the ramp, the G.P.E
would increase proportionally as we would see in graph three. And when
the G.P.E increases this means that when the object is moving
downwards the kinetic energy also increases, because there would be
more speed when rolling down. And that graph and results fully support
my prediction number two, which states that when Gravitational
potential energy increases the Kinetic energy also increases
proportionally.

Graph number three, shows how the G.P.E is affected by the height of
the ramp in centimeters. And the graph shows a positive gradient of
0.23J/cm this means that every centimeter you increase on the ramp
that I'm using the gravitational potential energy increases too by
0.23 joules. And that supports my prediction number three, which was
as the ramp height increases the G.P.E increases also proportionally.
Which also shows that the further away the object from the ground the
more G.P.E it's got stored in it, ready to be let out as Kinetic
energy.

Graph number four, after working out the gradient on the graph, shows
that as every one joule is added, would increase the speed of the ball
by 9m/s. which supports my prediction number four which states that as
the speed of the ball increase would mean that the kinetic energy
would also increase proportionally. And that also proves that if an
object wanted to move it would need a specific amount of kinetic
energy but if it wanted to go faster or stronger, it would need more
kinetic energy, that statement works both ways.

Graph number five, shows the proportionality between the friction
force and the kinetic energy. And the graph shows that at small amount
they are proportional but when they increase the friction becomes
bigger than the kinetic energy, in which that decreases the
efficiency, and loses energy as heats energy.

K.E *100 = percentage useful energy used

G.P.E

0.024051*100=24.051/0.188476=12.76% percent of the G.P.E transforms
into useful kinetic energy, the rest 87.24% is wasted as heat energy.
And that proves my last prediction which states that the friction
force is greater than the kinetic force!

Evaluating Your Evidence

During this experiment I have tested the relationships between the
height of a ramp and the speed of a ball rolling down it. Through this
I have come in to contact with the G.P.E, K.E. and the Friction
Forces. Which occurs relying on the variation in the height of the
ramp and the speed of the ball. All my predictions have been proved
using the graphs and the results, as my results were very accurate
except for a couple of anomalous results occurring. These anomalous
results may occur due to…:

Ø Timing the stop watch incorrectly, for example, letting the ball
roll first then pressing the start button on the stop watch a quarter
of a second later may alter the tests accuracy dramatically.

Ø The grooves on the ramp might have parts that decrease the speed of
the ball.

Ø The ball was not accelerating the whole time at the same
acceleration, which that would make a difference when it came to do
the calculations.

I don't think repeating any more test for results would make it more
accurate cause I've already had carried out 5 samples of each test.

I think I have carried out a suitable range of results but there was
an area for improvement as I could have investigated larger heights,
which would mean larger angles.

Improvements

Things that I would improve in this experiment are;

Ø Lager range of heights and angles

Ø Larger variety of ball to use

Ø A different surface for the ball to roll on, a smoother surface,
such as plastic, or metal, with would reduce friction

Ø A stop watch that is connected to the computer, and uses lasers to
start and stop the stopwatch when the ball rolls through them.

Ø Shorter ramp to keep the ball accelerating.

The whole experiment was a success in that I gained knowledge of
things I didn't know before, I have carried out the experiment safely,
fairly and accurately.