Introduction: In this investigation, how height will affect the time taken for a steel ball bearing to reach the ground will be investigated.

It is was Isaac Newton that first discovered gravity and wrote laws defining it. His Second Law of Motion states that the Resultant Force on an object (F) is equal to the Mass of the body (m) times its acceleration (a), or .

The weight (W) of a body is the force of gravity acting on it, which gives it acceleration (g) if it is falling freely close to the earths surface. If the body was to have a mass (m) Newtons 2nd Law of Motion could calculate its weight. Given that and Newtons Law becomes .

In April of 2003, in a method similar to that, which will be conducted in this investigation, the acceleration of gravity was concluded to be 9.81.

Using the knowledge mentioned above, several equation of motion have been created. One particularly relevant to this investigation is .

In this equation: S = Distance in meters (In this case height)

u = Initial velocity in *

t = Time Taken in seconds

a = Acceleration in **

* As the ball begins its fall from rest, its initial velocity, u, will be 0

** As the ball is falling under acceleration due to gravity, = 9.81

Plan: In this investigation, since the means to calculate air resistance and friction are unavailable, they will be ignored. Acceleration due to gravity and the method by which the investigation will be carried out are controlled variables. The height from which the ball is dropped is the independent variable, i.e. the variable that is changed and the dependent variable is the time taken for the ball to reach the ground.

It is predicted that as the height from which the ball is dropped decreases, the time taken for the ball to reach the ground will also decrease. An actual set of predicted values can be predicted from the earlier mention Equations of Motion.

This line of the equation can be related to the straight-line graph equation , (the gradient) and . When the ball is dropped from 0.00m, it takes 0.00seconds to reach the ground as this will mean that the graph will pass through the origin so .

Alternatively: .

Using the above equations, the expected results for the investigation are:

Height H, (m)

Time t, (secs)

Time squared t2, (secs2)

1.0

0.452

0.204

0.9

0.428

0.183

0.8

0.404

0.163

0.7

0.378

0.143

0.6

0.350

0.123

0.5

0.319

0.102

0.4

0.288

0.082

0.3

0.247

0.061

0.2

0.202

0.041

0.1

0.143

0.020

0.0

0.000

0.000

This is what the expected graph of Height vs. time should look like:

Height (m)

Time (secs)

The following page shows what the expected graph for Height vs. time2 should look like. It is a straight line passing through the origin, thus proving the prediction .

The expected gradient, m, should be equal to 1/2 g, or, 4.905ms-2. It is actually 4.926ms-2, which is only 0.021 ms-2 out or 0.428%. This is probably due to the rounding of decimal places when drawing the graph and human error in plotting the points (i.e. not exactly accurate to 3 decimal places.)

Apparatus & Diagram:

Safety: As there is a very minimal risk in this investigation, no safety measures need to be taken.

It is planned to drop the ball from a height of 1m and decrease in intervals of 0.1m. At each height 5 readings will be recorded and then the mean result will be calculated. This makes the results more reliable (and better for use in calculation like working out g or the mass of the steel ball.)

The Results will be recorded in a table like this:

Height (cm)

Time taken for ball to reach ground (seconds)

Mean Result

Mean Result2

1st

2nd

3rd

4th

5th

100

90

80

70

60

50

40

30

20

10

00

It is hoped that a graph of height vs. the mean results squared will be produced similar to that on page 4. In the graph, it is hoped to prove that the time-taken-for-a-ball2 to fall is directly proportional to the height it is dropped from, i.e. .

Obtaining Evidence:

Height (cm)

Time taken for ball to reach ground (seconds)

Mean Result

Mean Result2

1st

2nd

3rd

4th

5th

100

0.456

0.454

0.458

0.454

0.454

0.455

0.207

90

0.432

0.431

0.431

0.432

0.432

0.432

0.186

80

0.406

0.407

0.406

0.407

0.406

0.406

0.165

70

0.380

0.382

0.382

0.383

0.381

0.382

0.146

60

0.353

0.354

0.354

0.354

0.354

0.354

0.125

50

0.323

0.323

0.322

0.322

0.322

0.322

0.104

40

0.289

0.288

0.293

0.289

0.287

0.289

0.084

30

0.252

0.251

0.251

0.250

0.251

0.251

0.063

20

0.208

0.207

0.206

0.206

0.206

0.207

0.043

10

0.150

0.150

0.150

0.150

0.151

0.150

0.023

00

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Graphs: In the following pages, the results recorded in the above table will be shown in the form of line graphs. This will make it easier to identify a trend in the results. It is also an appropriate method of recording the information and is useful for quick reference; also if the time for the ball to fall is desired from a height other than the ones specified in the table, the value can be obtained from the graph.

Conclusion: The general trends from the graphs show, as predicted; when the height from which the ball was dropped decreased, the time taken for the ball to reach the ground also decreased.

In the graph of Height vs. Time2,, it is shown that Height is directly proportional to Time2. The reason for this is derived from one of the Equations of Motion: , from this equation below, it was shown that .

The final line of the above equation can be related to the straight-line graph equation . , (the gradient) and . C can be ignored as the line in the graph passes the y-axis at the orgin.

Fundamentally: .

H = S = The height in meters from which the ball was dropped.

= The time in seconds that the ball took to land.

a = g = The acceleration due to the gravitational pull of the earth.

Note: The factor that affected the acceleration was g, (which, on earth, is ) is the mass of the planet, for Earth this is constant.

The results of the investigation are consistent with the prediction. The relationship of was proved in the similarity of the graphs on page 4 & 8, they had almost the exact same gradient, only 0.072ms-2 in difference (or 1.462%) it was also very similar to the mathematical prediction of the gradient (1/2g) again only 0.051 ms-2 out.

Evaluation: In this investigation, all results are held to be very reliable. When the data was being collected, sophisticated technology was used which measured time accurately and reliably to the nearest thousandth of a second. All recorded results were in very close proximity of each other, so that 0.006seconds was the maximum difference observed.

There were no anomalies observed. All points on the graph on page 8 are not only close to the line of best fit, they are actually on it.

The results in this investigation are believed to be very reliable; as a result no changes need to be made to the procedure.

It is was Isaac Newton that first discovered gravity and wrote laws defining it. His Second Law of Motion states that the Resultant Force on an object (F) is equal to the Mass of the body (m) times its acceleration (a), or .

The weight (W) of a body is the force of gravity acting on it, which gives it acceleration (g) if it is falling freely close to the earths surface. If the body was to have a mass (m) Newtons 2nd Law of Motion could calculate its weight. Given that and Newtons Law becomes .

In April of 2003, in a method similar to that, which will be conducted in this investigation, the acceleration of gravity was concluded to be 9.81.

Using the knowledge mentioned above, several equation of motion have been created. One particularly relevant to this investigation is .

In this equation: S = Distance in meters (In this case height)

u = Initial velocity in *

t = Time Taken in seconds

a = Acceleration in **

* As the ball begins its fall from rest, its initial velocity, u, will be 0

** As the ball is falling under acceleration due to gravity, = 9.81

Plan: In this investigation, since the means to calculate air resistance and friction are unavailable, they will be ignored. Acceleration due to gravity and the method by which the investigation will be carried out are controlled variables. The height from which the ball is dropped is the independent variable, i.e. the variable that is changed and the dependent variable is the time taken for the ball to reach the ground.

It is predicted that as the height from which the ball is dropped decreases, the time taken for the ball to reach the ground will also decrease. An actual set of predicted values can be predicted from the earlier mention Equations of Motion.

This line of the equation can be related to the straight-line graph equation , (the gradient) and . When the ball is dropped from 0.00m, it takes 0.00seconds to reach the ground as this will mean that the graph will pass through the origin so .

Alternatively: .

Using the above equations, the expected results for the investigation are:

Height H, (m)

Time t, (secs)

Time squared t2, (secs2)

1.0

0.452

0.204

0.9

0.428

0.183

0.8

0.404

0.163

0.7

0.378

0.143

0.6

0.350

0.123

0.5

0.319

0.102

0.4

0.288

0.082

0.3

0.247

0.061

0.2

0.202

0.041

0.1

0.143

0.020

0.0

0.000

0.000

This is what the expected graph of Height vs. time should look like:

Height (m)

Time (secs)

The following page shows what the expected graph for Height vs. time2 should look like. It is a straight line passing through the origin, thus proving the prediction .

The expected gradient, m, should be equal to 1/2 g, or, 4.905ms-2. It is actually 4.926ms-2, which is only 0.021 ms-2 out or 0.428%. This is probably due to the rounding of decimal places when drawing the graph and human error in plotting the points (i.e. not exactly accurate to 3 decimal places.)

Apparatus & Diagram:

Safety: As there is a very minimal risk in this investigation, no safety measures need to be taken.

It is planned to drop the ball from a height of 1m and decrease in intervals of 0.1m. At each height 5 readings will be recorded and then the mean result will be calculated. This makes the results more reliable (and better for use in calculation like working out g or the mass of the steel ball.)

The Results will be recorded in a table like this:

Height (cm)

Time taken for ball to reach ground (seconds)

Mean Result

Mean Result2

1st

2nd

3rd

4th

5th

100

90

80

70

60

50

40

30

20

10

00

It is hoped that a graph of height vs. the mean results squared will be produced similar to that on page 4. In the graph, it is hoped to prove that the time-taken-for-a-ball2 to fall is directly proportional to the height it is dropped from, i.e. .

Obtaining Evidence:

Height (cm)

Time taken for ball to reach ground (seconds)

Mean Result

Mean Result2

1st

2nd

3rd

4th

5th

100

0.456

0.454

0.458

0.454

0.454

0.455

0.207

90

0.432

0.431

0.431

0.432

0.432

0.432

0.186

80

0.406

0.407

0.406

0.407

0.406

0.406

0.165

70

0.380

0.382

0.382

0.383

0.381

0.382

0.146

60

0.353

0.354

0.354

0.354

0.354

0.354

0.125

50

0.323

0.323

0.322

0.322

0.322

0.322

0.104

40

0.289

0.288

0.293

0.289

0.287

0.289

0.084

30

0.252

0.251

0.251

0.250

0.251

0.251

0.063

20

0.208

0.207

0.206

0.206

0.206

0.207

0.043

10

0.150

0.150

0.150

0.150

0.151

0.150

0.023

00

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Graphs: In the following pages, the results recorded in the above table will be shown in the form of line graphs. This will make it easier to identify a trend in the results. It is also an appropriate method of recording the information and is useful for quick reference; also if the time for the ball to fall is desired from a height other than the ones specified in the table, the value can be obtained from the graph.

Conclusion: The general trends from the graphs show, as predicted; when the height from which the ball was dropped decreased, the time taken for the ball to reach the ground also decreased.

In the graph of Height vs. Time2,, it is shown that Height is directly proportional to Time2. The reason for this is derived from one of the Equations of Motion: , from this equation below, it was shown that .

The final line of the above equation can be related to the straight-line graph equation . , (the gradient) and . C can be ignored as the line in the graph passes the y-axis at the orgin.

Fundamentally: .

H = S = The height in meters from which the ball was dropped.

= The time in seconds that the ball took to land.

a = g = The acceleration due to the gravitational pull of the earth.

Note: The factor that affected the acceleration was g, (which, on earth, is ) is the mass of the planet, for Earth this is constant.

The results of the investigation are consistent with the prediction. The relationship of was proved in the similarity of the graphs on page 4 & 8, they had almost the exact same gradient, only 0.072ms-2 in difference (or 1.462%) it was also very similar to the mathematical prediction of the gradient (1/2g) again only 0.051 ms-2 out.

Evaluation: In this investigation, all results are held to be very reliable. When the data was being collected, sophisticated technology was used which measured time accurately and reliably to the nearest thousandth of a second. All recorded results were in very close proximity of each other, so that 0.006seconds was the maximum difference observed.

There were no anomalies observed. All points on the graph on page 8 are not only close to the line of best fit, they are actually on it.

The results in this investigation are believed to be very reliable; as a result no changes need to be made to the procedure.