1. Chiu I Hsuan
Physics HL Year 1 Period 2
October, 20, 2010
The Angle of the Ramp Affects the Acceleration of the Ball Rolling Down the Ramp
Introduction:
When a ball is placed on an angled ramp, it accelerates down the ramp. The situation is
similar to skiing down a hill with no applied forces. The acceleration of the ball in affect of the
inclination angle of the ramp is investigated.
A force influences an object to undergo acceleration; there are mainly two types of forces, applied
force and conservative force. A common conservative force is gravity which is the main force involved in
this research. The equation of force in relation with acceleration is described as below.
F=MA [Equation 1]
Where F is force, M is mass and A is the acceleration.
Y axis
FNormal
Fgx
X axis
Fgy Fgravity
Ѳ
Figure 1: Shows a force diagram indicating possible forces acting on the ball in the situation.
According to Figure1, the net force acting on the ball in X axis is Fgx. Fgx = Sin (θ)Fg. If
combined with equation 1, the result equation is shown below. (Hecht)

2. A = g*Sin (θ) [Equation 2]
Where A is acceleration, g is the gravity and θ is the angle of the ramp to the ground.
Equation 2 indicates a proportional fit between A and Sin (θ), thus A is proportional to Sin (θ).
The result graph is expected to be a direct linear proportional fit which passes through the origin.
Design:
Research Questions:
How does the inclination angle of the ramp affect the acceleration of the ball rolling
down the ramp?
Variables:
The independent variable is the angle of the ramp to the ground, and the dependent
variable is the acceleration of the ball rolling down the ramp. The range of the independent
variable is 10.83degrees to 49.75degrees. The controlled factors are the temperature of the room,
dimensions of the ball used and the width of the U ramp. The temperature can affect data
collection for motion detector, because the motion detector requires the speed of sound in the air
in order to calculate the change in position over time. The temperature of the room was kept
constant by turning on the air conditioner. The same ball was used during the research, therefore
the size and the mass of the ball is kept constant. Since the wider the U ramp is, the lower the
acceleration, the width of the U ramp was kept constant.
Diameter of the Squash Ball: 5 centimeters
Width of the U Ramp: 1.5 centimeter
Figure 2: Showing the cross section view of the squash ball rolling down the U ramp.

3. Motion Detector
Squash Ball
Wooden Block
U Ramp
Figure 3: Demonstrates the set up of the investigation.
Hypotenuse
Height
Length
Figure 4: Naming of the ramp set up. Angle of the Ramp to the Ground (θ)
Procedure:
A motion detector was attached to a U ramp with tape. The temperature of the
LoggerPro set up for the motion detector was 26Celsius and the sample collection rate was
40samples per second. Wooden blocks were stacked under to vary the height of the ramp. The
motion detector was faced along the ramp, and after the data collection started; the squash ball
was set and released on the ramp, 20centimeter away from the motion detector. Data was
collected and saved. The setting of the derivative calculations was set to five data points. Then
the hypotenuse and the length were measured to calculate the angle of the ramp. The intervals of
the angles of the ramp to the ground are 10.83, 14.79, 13.13, 17.97, 21.65, 28.12, and
49.75degrees, changed by varying the number of wooden blocks under the ramp.

4. Data Collecting and Processing:
Angle of the Ramp and the Average Acceleration
Acceleration (±.09m/s2)
Angle of
the Sine
Length of the Height of the Ramp(θ) Angle Average
Ramp(±.001meter) Ramp(±.001meter) (±.2 ) (±.003) Trial 1 Trial 2 Trial 3 Acceleration
.713 0.99 1.01 1.07 1.02
.134 10.83 .188
.713 1.31 1.30 1.30 1.30
.162 13.13 .227
.713 1.66 1.47 1.51 1.55
.182 14.79 .255
.713 1.84 1.82 1.85 1.84
.220 17.97 .309
.713 2.16 2.19 2.19 2.18
.263 21.65 .369
.713 2.83 2.84 2.75 2.81
.336 28.12 .471
.642 4.92 4.84 4.80 4.85
.490 49.75 .763
Table 1: Presents the actual measurements of the ramp and the average acceleration for each
angle. The average acceleration is calculated by half of range of 14.79 .Data from angle 49.75
is used for sample calculation. The length and height of the ramp are shown in Figure 4.
Sample Calculation:
Sample Graph:

5. Figure 5: Demonstrating a raw data graph . The last two plots of the
acceleration trend were not included in the fit because the derivative calculation was set to use 5
data points to calculate the middle point. Therefore the last two was left out of the fit to prevent
unsuitable data.
Angle of the Ramp (θ):
Sin (θ) = (Height / Hypotenuse)
θ = Sin-1 (Height / Hypotenuse)
Sin (θ) = (.490 / .642)
θ = Sin-1 (.490 / .642)
θ = 49.75
Uncertainty of the Angle of the Ramp (θ):
Average: θ = Sin-1 (Height / Hypotenuse)
Maximum: θ = Sin-1 (Height + Uncertainty / Hypotenuse - Uncertainty)
Minimum: θ = Sin-1 (Height - Uncertainty / Hypotenuse + Uncertainty)
Uncertainty: (Maximum – Minimum) / 2
Average: θ = Sin-1 (.490 / .642)
θ = 49.75
Maximum: θ = Sin-1 (.490 +.001 / .642 - .001)
θ = 49.995
Minimum: θ = Sin-1 (.490 -.001 / .642 + .001)
θ = 49.508
Uncertainty: (49.995 – 49.508) / 2 = ±.2435

6. Sine Angle:
Sin (θ) = (Height / Hypotenuse)
Sin (θ) = (.490 / .642)
= .7632
Uncertainty of the Sine Angle:
Average: Sin (θ) = (Height / Hypotenuse)
Maximum: Sin (θ) = (Height + Uncertainty / Hypotenuse - Uncertainty)
Minimum: Sin (θ) = (Height- Uncertainty / Hypotenuse + Uncertainty)
Uncertainty: (Maximum – Minimum) / 2
Average: Sin (θ) = (.490 / .642)
Sin (θ) = .7632
Maximum: Sin (θ) = (.490 + .001 / .642 - .001)
Sin (θ) = .7659
Minimum: Sin (θ) = (.490 - .001 / .642 + .001)
Sin (θ) = .7605
Uncertainty: (.7659 – .7605) / 2
= ±.0027

7. Result Graphs:
Inclination Angle of the Ramp vs. Average Acceleration of the Ball Rolling Down the Ramp
Figure 6: Indicating a linear proportional fit between angle of the ramp and the average
acceleration.
Figure 7: Demonstrating a high-low fit of the ramp angle vs. average acceleration graph. The
uncertainty of the slope is ±.003, and the uncertainty of the y intercept is ±.04m/s/s

8. Sine of the Inclination Angle of the Ramp vs. Average Acceleration of the Ball Rolling
Down the Ramp
Figure 8: Indicting a proportional fit between sine of the angle and the average acceleration.
Note that the proportional fit does not go through all the plots.
Figure 9: Demonstrating a high-low fit of the sine angle vs. average acceleration graph. The
uncertainty of the slope is ±.3, and the uncertainty of the y intercept is ±.1m/s/s.

9. Conclusion:
Am equation is extracted as below from angle vs. acceleration result graph.
Acceleration = (.097±.003m/s2 ) Angle + (.06±.04m/s2) [Equation 3]
The equation demonstrates that the angle of the inclined ramp has a direct linear
relationship to the acceleration of the ball rolling down the ramp. No direct theory was used to
support this fit. This equation can be used to answer the research question.
The following equation is presented from a Sin (angle) vs. acceleration graph, derived
graph from angle vs. acceleration graph in order to compare the data with the theory mentioned
in Equation 2.
Acceleration = (6.1±.3m/s2)*Sin (Angle) + (0.0±.1m/s2) [Equation 4]
The slope of the equation refers to the acceleration of gravity in Equation 2. But a
considerable amount of energy was used in the rolling motion of the ball, therefore causes a
decrease in the acceleration. The theory does support Equation 4, but the value of the slope has
no precise evidence that indicates to the gravity. Because the trend was supported by the theory,
the data collected was also supported. Since Equation 3 and 4 came from the same set of data,
only derived, the data of acceleration vs. angle was also back upped by the theory.
The level of confidence is medium, because no theoretical support was given to the fit of
Equation 3, only the quality of the data collected was supported. The uncertainty of Equation 3 is
fairly small compare to its values, therefore gives a high level of confidence in data collection.
The fit in Equation 4 does not pass through all the data points, in effect; the fit of Equation 4 has
a low level of confidence.
The experiment result is applicable when a ball of the same nature and dimensions is
used on a U ramp with the same width in a rolling motion. Although not strongly supported, the
result of this investigation indicates a proportional relationship between the sine angle of the
ramp and the acceleration of the ball down the ramp in any similar situations. But the result
cannot be used when the ramp is inclined approximately over 50degrees.

10. Evaluation:
The method of the experiment has a few defects that might affect the result; such assuming the
rolling rate of the ball stays constant throughout all the angles, short data collecting time, and the possible
effect of air resistance that might built up in higher angles.
The less the ball rolls when travelling down the U ramp, the higher the acceleration, because it does
not use as much energy in the rolling motion. This can be improved by using a small block of metal so
that it slides instead of rolling.
Not enough sample points were collected during trials, because the ramp was too short and when the
angle higher, the average velocity of the ball increases, therefore causing difficulties in collecting
qualitative data. This can be improved by using a longer ramp, thus increasing the sample collecting time.
, air friction might cause
When a ball is rolling down a relatively high angle, in this case 49.75
an effect of decreasing the ball’s acceleration, because a hollow, light squash ball was used in the
experiment. The error can be eliminated by using a denser object, such as a small block of metal
as mentioned above.
An extension of this investigation would be researching the amount of force used in the rolling
motion of the ball. Thus the acceleration of a none rolling object is required to be collected and compared
to the acceleration of a rolling ball at each angle.