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# Modeling a basketball shoot in the lab

Extracts from this document...

Introduction

Modelling and investigating the farthest range from which a basketball can be shot into a ring

By Janice Lau (U6th)

Content Page

Aim

Background Information

Calculations and Diagram- prove that it’s a parabola

Theory- PROJECTILE MOTION AT AN ANGLE

How to model a basketball shot?

Apparatus

Force vs. Compression – Spring Loaded Plunger

Prediction/Safety

Experiment 1 - Preliminary investigation

Experiment 2

Experiment 3

Experiment 4

Experiment 5

Conclusion

Evaluation

Source

3

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6

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6-10

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11-12

13-14

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15-17

17-18

19-20

20-21

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The AIM of my investigation is to find the optimum angle for the maximum range for a basketball shot by modeling it in the lab.

Background- PROJECTILE MOTION

Definition: “An object launched into space without motive power of its own is called a projectile. If we neglect air resistance, the only force acting on a projectile is its weight, which causes its path to deviate from a straight line.”1

The projectile has a constant horizontal and vertical velocity that changes uniformly when it is influence by acceleration and gravity.

Diagram:

Fig 1 &2 shows that basketball shots are projectile motions, however, how can we show it mathematically?

Calculations

Consider the horizontal and vertical motion individually. Initially,

Ux = u cos θ ----- (1)

Uy = u sin θ----- (2)

The horizontal velocity is constant through out the motion, since the acceleration is vertically downwards. At time t, the velocity components are

Vx = Ux = u cos θ ----- (3)

Vy = Uy – gt = u sin θ –gt ----- (4)

The horizontal and vertical displacements of the objects are:

X= Uxt = ut cosθ ----- (5)

Y= Uy – ½ gt2 = ut sin θ – ½ gt2 ----- (6)

From equation (5) we have,

t= x/ u cos θ

Putting t into equation (6), the equation of trajectory is,

y= x tan θ – g/2u2 cos2 θ

Middle

Precautions/ Safety:

Before the experiment, I handed in a plan and was approved at a safety angle by teacher. During the experiment I have also considered many aspects of safety issues. For example, making sure that when I shoot the ball with the spring loaded plunger, I would not hit anyone and the use of balls; making sure the ball is not hard to hurt anyone.

Error in Range and % error calculation

Error in range = (max range – minimum range)/2

% error calculation = Error in range / Average Range

The following graphs’ error bars are plotted using the average percentage error.

Preliminary Investigations

In this investigation, a table tennis ball is chosen for trials because it is very light and is able to be fired off to a long distance.

Method:

I shoot a table tennis ball towards a sand pit at different angles using a “small” force and measure the range of the ball when it first land using a tape measure. I then record the corresponding horizontal distance as the angle varies; where angle is measure using a protractor. (Apparatus set up based on Fig 3.- without light gate)

Prediction:

I expect that the optimum angle for maximum range will be around 40-50 degrees, according to the theoretical value 45 degrees I have calculated earlier.

Results:

Diameter of the table tennis ball: 29 mm         Average weight of the ball: 2.6 g

 Angle(Degree) Range/ Horizontal Distance (m) 1 2 3 4 5 Average Error in range % Error in range 20 1.10 1.15 1.10 1.12 1.08 1.11 0.03 3 25 1.15 1.10 1.18 1.12 1.12 1.13 0.04 3 30 1.17 1.13 1.15 1.14 1.16 1.15 0.02 2 35 1.13 1.14 1.20 1.20 1.09 1.15 0.05 5 40 1.03 1.06 1.07 1.01 1.14 1.06 0.06 6 45 1.05 1.10 0.94 0.99 1.07 1.03 0.08 8

Table2. Shooting a table tennis ball with a ‘small force’- (for experiment 1)

Average error in range= 0.05 = 5%

Graph 2: Angles vs. Horizontal Distance- (for experiment 1)

Conclusion

5

41±5

"Big" force + squash ball+ sand pit at the same level as the shooting point

Conclusion

As the height of sand pit increases to level with launch, with a constant shooting point, the optimum angle increases and will be closer to the theoretical angle of 45 degree as it get to the same level as the shooting point.
From the results above, I can predict and expect that when the height of the sand pit is fixed (a basketball ring- in real life), the optimum angle for a lower shooting point (short person) will be further from the optimum angle than from a higher shooting point (a taller person).

Evaluation

Although my experiments proved the trend between height and optimum angles, there are lots of other ways to further investigate and improve my investigation.

There are errors in measurements

• Apparatus error e.g. the accuracy of protractor and ruler for measurements is ±0.5 cm and ±0.5 degree respectively.
• Inconsistency of the force when shooting using a spring loaded plunger

To improve my investigation, I can firstly use more accurate apparatus. Then have a better and wider range of results with a smaller interval between results.

For further investigations, I then find out the relationship between different types of balls and different speed of balls and see how they affect the optimum angle of a shot.

Sources

1. http://apcentral.collegeboard.com/apc/members/repository/ap03_projectile_motio_30832.pdf (reference for definition)- page1
2. http://www.lightingsciences.ca/pdf/BNEWSEM2.PDF
3. Advanced Physics (p.118) by Tom Duncan
4. Mechanics 2 by John Hebborn and Jean Littlewood
5. http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/ProjectileMotion/jarapplet.html
6. http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/30827.html
8. http://apcentral.collegeboard.com/apc/members/repository/ap03_projectile_motio_30832.pdf

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