To find the optimum release height for scoring
PART ONE Study the relationship between the possibility of scoring and the distance between the person and basket
Since we assume that the person has infinite strength, and the basketball can be thrown as high as possible, it can be concluded that the height of the board determines the vertical wiggle room of all situations.
From these two graphs, we can clearly find that by adjusting the trajectory, the basketball can hit anywhere on the board to make a bank shot.
However, if the distance changes, the horizontal wiggle room will also change.
However, if the distance changes, the horizontal wiggle room will also change.
PART TWO Study the relationship between the relationship between the possibility of scoring and the angle between zero degree line and point of shooting
The farther the player is, the less possibility of scoring will be.
In this case, we still measure the wiggle room to determine the possibility of scoring.
l is the distance from the board to the center of the basket
p is the difference between the radii of the basket and the basketball.
x is the distance from the player to the center of the symmetrical basket.
Ө is the angle between zero degree line and point of shooting
After substituting the first function into the second function,
the difference between two definite integrals indicates the possibility of scoring.
p is the difference between the radii of the basket and the basketball.
x is the distance from the player to the center of the symmetrical basket.
Ө is the angle between zero degree line and point of shooting
After substituting the first function into the second function,
the difference between two definite integrals indicates the possibility of scoring.
At the ninety degree line, the possibility of scoring is the largest.
2. The relationship between s and angles
the green circle is the reflection of the basket about the board.
We decides to define the position of the player on the board with two variables.
The distance to the center of the basket (x HO) and the angle with respect to the center of the basket (θ)
We believe that the possibility of scoring is proportional to the possible distance of scoring (FG) and the possible angle variation (γ)
We decides to define the position of the player on the board with two variables.
The distance to the center of the basket (x HO) and the angle with respect to the center of the basket (θ)
We believe that the possibility of scoring is proportional to the possible distance of scoring (FG) and the possible angle variation (γ)
Let us define
By using geometry knowledge and trigonometry,
we can calculate these function
we can calculate these function
By using the software, we decides to isolate θ first, to study what influence does θ have on s.
let x1=0.425(according to real life data)
r=0.23
x=6
Geometry Approach
Please focus on the data on the right corner
We can see when θ∈ (0,π/8), there is max(s)
[this is to show how to find the maximum s using geometry's software,
the scale of the graph is disproportional due to its use for illustration
Therefore the data is invalid ]
r=0.23
x=6
Geometry Approach
Please focus on the data on the right corner
We can see when θ∈ (0,π/8), there is max(s)
[this is to show how to find the maximum s using geometry's software,
the scale of the graph is disproportional due to its use for illustration
Therefore the data is invalid ]
Function Approach
(Using Desmos Graphing Calculator)
This time we plug in the real data and the graph is more accurate.
(Using Desmos Graphing Calculator)
This time we plug in the real data and the graph is more accurate.
Plot the above graphs
This is what we get, the blue stands for f(x), green g(x), purple h(x)
We can see the maximum point of the graph h(x) for x∈ (0,π/2) is near x=0, the difference is almost negligible.
Our suggestion about the reason of the difference between the geometry approach and function approach is the difference of the data taken for the basket. If we increase the radius of the basket, we can see the max(s) happens not exactly when θ equals to 0, but a little more than 0.
Considering the radius of the basket is almost negligible as compared to the distance of player to the board in the real life, that is why the difference in the graph is negligible.
Our suggestion about the reason of the difference between the geometry approach and function approach is the difference of the data taken for the basket. If we increase the radius of the basket, we can see the max(s) happens not exactly when θ equals to 0, but a little more than 0.
Considering the radius of the basket is almost negligible as compared to the distance of player to the board in the real life, that is why the difference in the graph is negligible.
Next we isolate the variable γ.
From the geometry approach, we can see γ has its maximum when θ equals π/2.
From the second formula, we can see the γ is larger, d is smaller
From the geometry approach, we can see γ has its maximum when θ equals π/2.
From the second formula, we can see the γ is larger, d is smaller
According to the cosine rule, we can see if θ is larger,d is smaller and γ will be larger.
Therefore γ has its maximum when θ equals π/2.
That is the algebra approach.
d is smaller when θ is 0, according to two sides of the triangle is always more than the other side(DJ=DO+OH>DH)
That is the geometry approach.
Analysis
Strangely, we find s reaches its maximum when θ is 0, γ reaches maximum when θ equals π/2.
Considering that both factors influence the possibility of scoring,
the best θ to score maybe somewhere near when θ equals π/4.
Further calculation is needed.
Therefore γ has its maximum when θ equals π/2.
That is the algebra approach.
d is smaller when θ is 0, according to two sides of the triangle is always more than the other side(DJ=DO+OH>DH)
That is the geometry approach.
Analysis
Strangely, we find s reaches its maximum when θ is 0, γ reaches maximum when θ equals π/2.
Considering that both factors influence the possibility of scoring,
the best θ to score maybe somewhere near when θ equals π/4.
Further calculation is needed.