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Scoring a Basket.

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Introduction

MATHS COURSEWORK – SCORING A BASKET

Task: Find values for the initial speed and angle of projection needed to score a basket in basketball from the free throw line. Is there a range of values of angle for an initial velocity that will still allow you to score a basket?

Method: To solve this problem I am going to record myself scoring 10 clean baskets (the ball does not touch the hoop or back board) on video camera. Having done this I will know the time taken from the ball leaving my hands to it going into the basket. By using vector equations for Displacement, Velocity and Aceeleration in relation to time I will be able to calculate:

  • The maximum height achieved by the ball.
  • The distance travelled by the ball
  • The time taken to travel this distance
  • The height of release and the height at which it lands.

This will give me an accurate model of the balls motion through the air. I will be able to use the velocity vector equations to find the horizontal and vertical components of velocity for each throw. Using Pythagoras Theorem I will be able to calculate the initial velocity. Using trigonometry I will also be able to calculate the angle of release. This will allow me to analyse how initial velocity and the angle of release are related and therefore allow me to find a range of values for the angles of release for a given velocity.

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Middle

3

1.03

1.950

54.339

7.614

4

1.01

1.903

53.468

7.605

5

1.1

2.125

57.232

7.679

6

1.07

2.049

56.021

7.645

7

1.16

2.286

59.521

7.770

8

1.12

2.178

58.014

7.706

9

1.06

2.024

55.608

7.636

10

1.09

2.099

56.833

7.667

Average

1.077

2.070

56.252

7.663

I have also investigated the velocity needed to achieve a basket for any given angle. Although I dont know the values for the horizontal and vertical components of velocity they can be represented by

[             ]

These values can be put into the position vector equation to give:

[                            ]

I know that the horizontal component is equal to 4.572 m so I can rearrange to give:

This can then be substituted into the vertical component of the position equation. This leaves only t as an unknown quantity so:

This gives the total time for the throw. This value can now be substituted into the velocity vector equation to find the initial velocity as shown earlier.

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Conclusion

        Finally, the experiment in general had the possibility for a lot of errors. For example there was human error involved in the timing of each throw and the measuring of the release point so the answers derived from these values would be inaccurate. Also because there are so many calculations it means that any error gets increasing multiplied to cause an even bigger inaccuracy. Having said this there isn’t a lot that can be done to gain more accurate results without using advanced equipment. For example light gates could be used to measure the time of the throw more accurately and machines could be used to throw the ball so that the release height, velocity and angle could all be better controlled.  

There are a number of things that could be investigated to achieve a more accurate representation of the angles needed to score a basket. Firstly I only looked at the angle of release for one release height. If the release height was higher or lower then the release angle could change or the velocity needed for a certain angle would be different which might make the optimum angle for different people. Secondly you could look at the affect of jump shots from the free throw line, again this is increasing the release height.  

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