PROJECTILE MOTION

NAME:

STUDENT NUMBER:

SESSION TIME:

APPARATUS REQUIRED PER GROUP

1 Metre Rule
1 Curved metal track
2 Retort stands
2 Boss-heads
2 Clamps
1 Ball bearing
1 Plumb bob
1 Spirit level
2 A3 sheets of plain paper
1 A3 sheet of carbon paper

When finished using the equipment return it NEATLY back to where you found it.
THIS INCLUDES THE CARBON PAPER. IT IS REUSABLE
Include the plain paper with your results.

SAFETY ISSUES

Always wear closed shoes to protect your feet from injury.

A lab coat is mandatory in all generic laboratories.
Take care when moving retort stands
BACKGROUND AND THEORY

In this experiment a ball bearing will be released from rest and allowed to follow a curved path
before it exits of the end of a bench with a horizontal velocity, 𝑣!" (Figure 1).
The horizontal velocity can be determined using the conservation of energy relation:
Loss of potential energy = gain in kinetic energy + gain in rotational energy
The horizontal velocity is found, assuming no energy losses due to friction to be:

10𝑔ℎ
𝑣!" = $ 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 1
9

Upon leaving the bench, the ball bearing will follow a parabolic path. Galileo demonstrated
through experiment and deductive logic that the horizontal and vertical components of parabolic
motion can be treated separately.
Assuming the bearing leaves the bench horizontally and hence with zero velocity in the vertical
direction the motion in the horizontal direction can be found using an equation of motion given
by:
𝑥 = 𝑣!" 𝑡 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2
Where 𝑥 is the distance travelled by the projectile (ball bearing) in the horizontal direction, and 𝑡
is the time of flight of the projectile.
The distance fallen in the y-direction is found by a second equation of motion that describes
accelerated motion:

𝑦 = 152 𝑔𝑡 # 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3
Galileo was the first to show that these two equations still work when an object is moving
horizontally and falling at the same time. This fact allows us to combine them to get an equation
relating 𝑦 and 𝑥, without 𝑡 appearing in the equation.
Equation 2 (horizontal motion) can be rewritten as:
𝑡 = 𝑥5𝑣!"
Substituting this expression for 𝑡 into equation 3 (vertically accelerated motion) we obtain:
#
𝑦 = 152 𝑔 𝑥 5𝑣 # 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 4
!"

Hence the path traversed by the projectile is parabolic.

Rearranging equation 4 we finally obtain:
#
2𝑣!"5𝑔9 𝑦
𝑥# = 8 𝑜𝑟 𝑥 # = 𝑘𝑦
#
2𝑣!"5𝑔
Where, 𝑘 =

Hence by measuring 𝑥 and 𝑦 and plotting a graph of 𝑥 # versus 𝑦 you will obtain a straight line
with a gradient given by 𝑘.
Since 𝑘 is associated with the horizontal velocity its value can be determined and compared with
the theoretical expression given by equation 1.

OBJECTIVES

1. To confirm that the relationship between the 𝑥 and 𝑦 displacements of an object

undergoing projectile motion is non-linear.
2. To measure the horizontal component of velocity and compare the measured value with
the theoretical value derived from energy conservation.

PROCEDURE
1. Assemble the components according to the schematic diagram shown in Figure 1.
Sandwich the carbon paper between two sheets of plain paper. Ensure that the carbon
paper goes carbon side down. When the ball bearing lands on the top paper it will leave a
mark on the bottom paper. Secure the paper combination to the table using tape.

Track
h
Bossheads
& Clamps
Vox

Ball

Retort
Stands Paper y

Paper Carbon
Bob Paper

Figure 1. Schematic diagram of the projectile motion experiment

2. The launch track is constructed of aluminium and is bent into a quarter circle and
mounted onto a wooden platform. Figure 2 shows how you can mount the track using
clamps and boss-heads to the retort stands. It is important that the ball leaves the track
horizontally so that there is no initial vertical velocity. Use the spirit level to ensure that
the track is horizontal when the ball exits.
 bossheads

retort
stands
clamps

track

Figure 2. Mounting configuration of the launch track

3. Attach the plumb bob to the front edge of the launch track. All measurements of 𝑥 will
be taken from the bob to the marks made on the paper by the ball bearing when it lands.
4. Always release the bearing from the same height, ℎ.
5. Release the ball bearing from ℎ on the track ten times for each value of 𝒚 that you
test. As a result, you will obtain a group of marks where the ball lands (Figure 3).
6. Measure the minimum, 𝑥$%& and maximum, 𝑥$'" values. Determine the uncertainty, ∆𝑥
("!"# # "!$% )
using . In the example shown in Figure 3, this results in an uncertainty of
&
(41.8 – 39.8)/2 = 1cm. Record the uncertainties in Table 1.

4 1.8cm
39.8cm
Point where the ball 2.0cm
exits ramp, indicated
by plunge line

x cm

Figure 3. A sample set of results.

RESULTS
The height ‘ℎ′ of the track is ___________ ± __________ cm
#
𝑦 𝑥$%& 𝑥$'" 𝑥'()*'+) ∆𝑥 𝑥'()*'+) ∆𝑥 #
±_____cm (cm) (cm) (cm) (cm) (cm2) (cm2)

Table 1. Vertical height ‘𝑦’ and corresponding ‘𝑥’ measurements and associated uncertainties
ANALYSIS
1. Using your data in Table 1 plot 𝑥 # on the y-axis and 𝑦 on the x-axis. Include error bars
for each measurement. See overpage for graph paper.
2. Draw a line of best fit, LBF. Determine the gradient of LBF.
This is represented by 𝑘,
Enter your calculation of the gradient in the box provided below.

3. Determine the gradient of the line of worst fit (upper), 𝑘-

4. Determine the gradient of the line of worst fit (lower), 𝑘.

 Plot of 𝑥 # against 𝑦 for a projectile
5. Calculate 𝑣!"! , 𝑣!"" and 𝑣!"# using your 𝑘, , 𝑘- and 𝑘. values.
6. Calculate the difference between the upper fit and best fit: ∆𝑣!"$ = A𝑣!"" − 𝑣!"! A

7. Calculate the difference between the best fit and lower: ∆𝑣!"% = A𝑣!"! − 𝑣!"# A

8. Whichever value is larger, ∆𝑣!"$ or ∆𝑣!"% is the uncertainty in the velocity.

State your final value of velocity and its uncertainty as:

𝑣!" = ________________ ± ____________ 𝑚⁄𝑠

9. Calculate the expected value for 𝑣!" using Equation 1

10. Calculate the percentage error in your measured value.

DISCUSSION

Include in your discussion:

1. A justification about why the uncertainty in the measurement of 𝑦 is likely to be greater

than the limit of reading of the instrument.
2. Why linearising the equation proved to be a useful technique.
3. The experimental factors that have an impact on the value of 𝑣!" and how these factors
directly affect the value of 𝑘.
4. A reason, therefore, why the line of best fit should not be forced through the origin.
5. A statement about the accuracy of your results using the percentage error calculation –
compare your value with the accepted value.
6. A statement about the precision of your results. Refer to your final uncertainty.
CONCLUSION

Include in your conclusion:

1. A statement given your value of horizontal velocity and its uncertainty.

2. A statement about the validity of the methodology used to support the model
(equation 1) based on the accuracy and precision of your measurements.