Basketball: Will You Bank the Shot?
AbstractYou are right next to the basket and someone passes you the ball. Will you go for a direct shot or will you use the backboard and take a bank shot at the basket? Would different positions on the court give you a higher chance of making a shot using the backboard than others, even when keeping the distance from the hoop the same?
In this science project, you will build a scale model and test different positions on the court to determine if one results in a better chance of making a bank shot than others.
ObjectiveCreate a two-dimensional scale model to determine the relative chance of scoring a basket using the backboard from different positions on the court.
With a tied-score and seconds to go on the clock who wins comes down to both who has the ball and where they are on the court. Even with the ball in hand it feels like some shots are harder to make than others—but is there science behind that gut feeling?
Figure 1. The concentration on this basketball player's face is clear. Whether or not she makes the basket though could have something to do with where on the court she is shooting from. Do you think that different court locations give you different odds of making a banked shot? (Photo courtesy of U.S. Air Force / J. Rachel Spencer, 2009)
As you probably know, basketball players have several shooting techniques to choose from, like straight shots, where the ball enters the basket without touching the backboard, or banked shots, where the ball bounces off of the backboard before it enters the basket. They often have to make a split-second decision. Should a player decide based on his or her position on the court? Would it be easier to make a banked shot from, let us say, the point guard position (position 1 in Figure 2), the shooting guard position (position 2 in Figure 2) or the small forward position (position 3 in Figure 2)?
Figure 2. Different positions on the basketball court might result in a relatively higher or lower shooting success for banked shots.
The NBA (National Basketball Association) keeps shot charts of actual games, and scientists use computer-based models to analyze which positions provide higher rates of successful shots. In this science project, you will build your own two-dimensional scale model to see whether different positions on the court result in a higher chance of making a successful banked shot.
In real basketball, the ball follows a complicated three-dimensional path (also called a trajectory) as it moves through the air, bounces off the backboard, and goes into the net. However, it is an example of a compound motion, combining a vertical and a horizontal movement. Both these motions are independent of each other and thus, can be studied separately. In this science project, you will study the horizontal movement of the basketball, which reduces the three-dimensional motion to two dimensions. This makes it easier to analyze for your science project. It is like having a bird's-eye view of the game; you will only study how the ball moves horizontally. This will also allow you to study a ball that is rolled instead of thrown through the air in your scale model.
Are you surprised that the ball's vertical (up and down) and horizontal motions happen independently? You can experiment with this type of compound motion by letting identical objects (like two coins) fall to the ground using different paths (e.g. one coin is launched off the table while the other is nudged right off the edge and falls straight down from the table). See when both objects land. If vertical movement (falling straight to the ground) happens independent of the horizontal movement (going sideways), both coins should touch the ground at the same time, even though the first object took a longer path.
To isolate the horizontal movement, imagine looking at the court from a bird's-eye (top-down) view. Imagine a player is shooting a ball. What would you see? Can you visualize the trajectory of the ball being shot, bounced on the backboard and landing in the basket? Do you imagine a trajectory as shown in Figure 3?
Diagram of a bank shot in basketball. The diagram shows the path of a ball that bounces off the backboard before going through the basket. When shooting from an angle to the left of the basket, a player must aim to the left of the rim in order to make the bank shot.
Figure 3. Top-down view of the trajectory of a basketball falling into the basket after bouncing off of the backboard. The position of the player is defined by an angle (player's angular position) and a distance (player's distance).
While sports people refer to positions on the court with specific names, scientists use variables to define positions on a two-dimensional field, like a basketball court. You need two variables to define a position on a two-dimensional field. In this science project, you will define the position of your player by the player's distance from the center of the basket and the player's angular position, as shown in Figure 3.
Scientists test one variable at a time to see the effect this one variable has on the outcome. In this science project, you will study how changing the angular position of the player changes the chances of making a successful banked shot. As you can only study one variable at a time, you will need to keep the player at a constant distance from the center of the basket.
You will use relative probability to express your results in a scientific way. A relative probability indicates how much more or less likely something is expected to happen compared to a chosen baseline. Let us say you choose position 1 of Figure 2 (the player's angular position is 90°) as the baseline for comparison. In that case, your relative probability will express how much more difficult or how much easier it is to score a banked shot at a specific position with respect to position 1 (the chosen baseline position). A relative probability lower than 1 indicates it is less probable, or more difficult, to successfully shoot a banked shot from that position than it is from the baseline position. A relative probability higher than 1 indicates it is more probable, or easier, to shoot a successful bank shot from that position than it is from the baseline position.
Alright, time to create your own scale model, roll some balls, and have a critical look at your results. Do not forget to keep this science project in mind the next time you are making that split-second decision about whether or not to use the backboard to shoot a basket!
Terms and Concepts
- Banked shots
- Compound motion
- Relative probability
- What property allows analyzing the projectile motion in the horizontal plane and the vertical direction separately?
- How does the trajectory of a basketball shot look in the horizontal plane?
- What does a relative probability of 2 or .5 indicate?
- Why would you only change one variable (like the angular position), keeping the other variable(s) (like the distance to the hoop) constant in a science project?
- Do you expect a specific position on the court to provide a better chance to score using the backboard?
- Do you expect any players placed on the court at equal distance from the center of the basket, but at different angular positions, to have a better chance to score using the backboard?
- Silverberg, L.M.; Tran, C.M.; Adams, T.M. (2011). Money in the Bank: Using Backboard Can Pay Off for Basketball Shooters. NC State University Newsroom. Retrieved December 8, 2013.
- The Physics Classroom. (n.d.). Independence of Perpendicular Components of Motion. Retrieved November 24, 2013.
Materials and Equipment
- Test ball, approximately 4.5 centimeters (cm) diameter. A mini sports squeeze ball like the mini sports foam balls from Amazon.com work well.
- Note:You will need to scale the dimensions of the cardboard tube and the number of poster boards up or down to accommodate a smaller or larger ball.
- Books (2), identical thickness, not too thick
- Flat wall making a 90° angle with a flat floor
- Ruler, metric
- Poster boards (2), 22 x 28 inches; flexible or hard poster board will both work.
- Thick markers, black and red
- Sheet of paper
- Tape that does not damage the wall
- Cardboard tube (50 cm long if you work with a 4.5 cm ball ). The inner tube of packing paper works well. Note: Choose something that comes on a sturdy cardboard tube, not on a flimsy wrapping paper tube.
- Utility knife or craft knife
- Cutting board
- Toilet paper roll
- Lab notebook
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Defining the Variables
A position on a two-dimensional field is specified by two variables. In this science project, you will use a radial distance (the distance to the center of the basket) and an angle (the angle that the line through the player and the center of the basket makes with a line parallel to the backboard), as shown in Figure 4.
Top-down view of a diagram where three players stand at 30, 60 and 90 degrees from a basket that is 3 meters away. The 0 degree axis begins in the center of the basket and extends out towards the corners of the court.
Figure 4. The position of a player is defined by the player's distance from the center of the basket and his angular position. In this science project, players are positioned at a constant distance from the center of the basket and varying angular positions.
As mentioned in the Background information, it is important to study one variable at a time. For this science project, you will stick to one distance and study different angular positions, being the 0°, 30°, 60°, and 90° positions, as shown in Figure 4. A distance of 3 meters (m) will be used throughout the procedure. As a variation, you can replace this distance by your own preferred distance. (See Variations for details). Also note that metric units are used, as this is the convention for science projects.
You will determine how much easier or more difficult it is to bank a shot (relative probability) with respect to a baseline position. The 90° position is chosen as baseline for this science project, which means you will test how much more or less difficult it is to make the banked shot from the 0°, 30° and 60° positions with respect to the 90° position.
Creating a Scale Model
You will make your basketball court model to scale, which means it will have the same proportions as in the real-life game, only scaled down to the size of your chosen ball.The experiment is symmetrical on the left and right sides of the basket, so you only need to test one side. The scale model described here only considers test positions on, or to the left of, the basket.
- Determine how much smaller your scale model will be compared to the real game dimensions. To do this, you need to compare the diameter of your test ball to the official dimension of a basketball (9.4 inch diameter for a size 7 ball).
- To measure the diameter of your test ball expressed in centimeters, hold the ball against the wall with a book on the other side, as shown in Figure 5. The distance between the wall and the book provides an accurate measurement of the diameter of the ball. Note your measurement down in your lab notebook.
- The scale factor is the ratio of the diameter of the test ball to the diameter of a real basketball, or:
For a 4.5 centimeter (cm) diameter test ball, this ratio is 0.479 cm/inch. This means that every 0.479 cm in the scale model will correspond to 1 inch on the basketball court. Calculate the scale factor by filling in the diameter of your test ball expressed in centimeters into the equation. Note the result down in your lab notebook. Do not forget to add the units (centimeters or inches).
- Note that you need to divide by this scale factor to convert from your scale model to the real game. In other words, if you divide by the scale factor obtained in step 1.b., you see that every centimeter in your scale model corresponds to 1/0.479 or 2.09 inches in the real game.
- For this science project, you will also need to convert the scale factor that you calculated in centimeters or inches to metric units (centimeters or meters). Could you calculate how many centimeters in the scale model correspond to 1 m on the court? (Hint: You will need to express the diameter of a real basketball in meters and use Equation 1.)
Figure 5. Placing your test ball between a wall and a book allows you to accurately measure its diameter. The ball shown here has a diameter of approximately 4.5 cm.
- Calculate the dimensions of your scale model.
- Copy Table 1 in your lab notebook; it will allow you to easily keep track of all numbers and units.
- Note that this table has been filled in for a 4.5 cm diameter test ball. If you are using a test ball with a different diameter, recalculate all italic values in the third column by multiplying the distances on the real court expressed in inches (found in the second column of the table) by the conversion factor obtained in step 1.b.
- Remember that although official basketball dimensions are expressed in inches, scientists always use metric units. For your science project, you should measure distance in metric units.
|Official basketball dimensions||Scale model dimensions (cm)|
|Ball diameter||9.4 inches||4.5 cm|
|Basket diameter||18 inches||8.6 cm|
|Shortest distance from the edge of basket to the backboard||6 inches||2.9 cm|
|Backboard length||72 inches||34 cm|
|Player's distance to the center of the basket||118 inches or 3 m||56 cm|
- Create your scale model. Take one of the poster boards and draw a top-down view of the basket and backboard, scaled down to the size of your ball. Figure 6, will guide you through the process.
- Put the poster board lengthwise in front of you.
- In the table you made (like Table 1), check how long your scaled down backboard needs to be. If you are using a 4.5 cm test ball, the backboard length should be scaled down to 34 cm.
- Start about 20 cm from the right side of your poster board. Indicate this point with a small mark on the top edge of your poster board. Your basket will be placed at this spot.
- From the point you just drew, measure a distance one-half the length of your backboard to the left and to the right, and make new marks. These outer marks represent the edges of your backboard. Note: Do not worry if the right edge of the backboard falls just off the poster board; you will be testing the left side only.
- Make small marks, 1 cm apart, starting from the center of the backboard all the way to the left edge of the backboard. These marks will help identify the spot where the ball touches the backboard in your scale model.
- Refer to the table (like Table 1) to find the distance from the backboard to the nearest edge of the basket and draw a line to represent the metal that holds the basket on the backboard (red line in Figure 6). This line is perpendicular to the backboard (edge of the poster board). This line will be referred to as the central line.
- Measure the radius of the basket from the end of the central line drawn in the previous step. This point is the center of the basket.
- Now place the sharp part of the compass on that point and use it to draw the circular basket on your poster; make it well visible by going over it with a thick black marker.
Figure 6. Top-down view of the backboard, the metal attaching the basket to the backboard (red line), and the basket, drawn at the edge of a poster board.
- Indicate the player's directions on the poster board. You might want to start with a pencil line and accentuate part of this line with a marker if you wish to.
- To indicate the 0° direction, draw a pencil line parallel to the poster board edge (and thus parallel to the backboard) through the center of the basket.
- Use a protractor to indicate the 90°, 60°, and 30° directions by placing the center of the protractor in the middle of the basket and the flat line of your protractor parallel to the edge of your poster board. The 0° marks on your protractor should line up with the 0° direction you drew in step 4.a. Note: Make sure to measure the angle from the middle of the basket,—as shown in Figure 7—and not from the edge of your poster board.
Figure 7. The central point of a protractor is placed at the center of the basket to measure the 90°, 60°, and 30° directions. The player's position is measured from the center of the basket (0 cm mark of the ruler is placed at the center of the basket), as explained in step 5.
- Indicate the player's positions on the poster board. Note: You might need to attach poster boards (or pieces of paper) to the first poster board, as all of the player's positions might not fit on your first poster board.
- Find the scaled down player's distance to the center of the basket in the table you made similar to Table 1. For a 4.5 cm diameter test ball, the chosen distance of 3 m on the real-life court scales down to a distance of 56 cm.
- Put your ruler or measuring stick in the 0° direction, line up the 0 cm mark with the center of the basket and indicate the player's position at the correct distance (56 cm) with a marker.
- Repeat step 5.b. for the 30°, 60°, and 90° directions.
- Place your poster board on the floor near the wall, with the backboard drawing against the wall.
Create a scaled down backboard so you can easily identify the distance from the central line at which the ball touches the backboard. Let Figure 8, be your guide.
- Look up the scaled down length of your backboard in the table you made like Table 1. Divide this by 2, as you will only use the left side of the backboard in your test.
- On a separate piece of paper, draw parallel lines perpendicular to the edge of the paper, 1 cm apart and about 10 cm long, until you have enough to cover half of the scaled down backboard (i.e. 17 lines if you are using a 4.5 cm ball).
- Number your lines starting from 0 (rightmost line) toward 17 (the line farthest away from the basket).
- Tape this paper to the wall so the line marked "0" lines up with the central line. Make sure the tape you are using will not damage the wall.
Figure 8. A paper with parallel lines taped against the wall is used to better identify at which distance from the central line the ball hits the backboard (wall). The 0 cm mark identifies the central line.
- Create a ramp for the ball, representing the path the basketball takes from the player to the backboard. Note that from a bird's-eye view, the ball's trajectory follows a straight line.
- The ball rolling down the ramp will represent the approximate 3 m horizontal path the ball takes from the player to the backboard. If you are using a 4.5 cm ball, this path scales down to approximately 56 cm long. Cut your cardboard tube to about 50 cm long, leaving some space for the test ball to bounce back. Note: You can use scissors or a craft or utility knife for this and for the following step. Make sure to have an adult supervise or help you if you use the craft or utility knife.
- Cut your cardboard tube lengthwise. Try to cut in a straight line, parallel to the length of the tube. Do not cut it in half, just one straight line cut.
- Gently pry your tube open so it forms a long, U-shaped track, as shown in Figure 9.
- You will elevate the track a little so gravity makes the ball roll. Test your track a couple of times to determine the best elevation.
- Let the ball roll down the track and bounce off of the wall, holding the far end (representing the position of the player) at heights from 5 cm up to 11 cm.
- Determine which height gives a nice roll to the ball and bounce on the wall, but that doesn't cause it to roll too quickly.
- Cut a toilet paper tube so it matches the height determined in step 7.d. This tube will be used as a stand to slightly elevate the track, as shown in Figure 9. Feel free to decorate the tube so it looks like a player.
Experimental setup of a cardboard tube cut in half lengthwise to form a slide for a small basketball to roll down. The basketball is rolled towards a paper marked with measurements that allow a person to see where the ball makes contact with the wall. The ball then bounces off the wall and rolls over a basketball hoop that is drawn on a separate piece of paper on the ground. The paper on the ground includes angles measured so that the tube can be placed at various angles around the basket.
Figure 9. A cut-open cardboard tube guides the ball toward the scale model backboard. A toilet paper tube cut in half is placed on the player position to give the track a small slope. Note that although the ramp starts from the 30° position, it does not necessarily follow the 30° degree line; it would only do that if the track was aimed at the middle of the basket. But since you are testing a banked shot for your trials, the track is aimed so the ball will impact the backboard at a specified distance from the central line.
- Evaluate if the data taken from the 0° position is realistic in your scale model.
- Depending on your model, it might be very hard to collect data for very small angles. If you cannot collect data for the 0° angle, add a player's position at a small angle (such as 10°) by repeating steps 4.b. and 5.b. for the chosen small angle. In the data table like Table 2, introduced in the next step, write whatever information you can get for the 0° angle and add a column for the player's position at the small angle you added.
- Before you start your trials, copy the table in your lab notebook to record your results.
|Ball Impact Position from the Central Line on a Real Court (cm)||Ball Impact Position from the Central Line in the Scale Model (cm)||90° position||60° position||30° position||0° position|
|Total successful banked shots|
Table 2. Table in which to record how many of 10 trials resulted in a successful banked shot. "Ball impact position" refers to how far from the central line the ball hits the backboard. The leftmost column is for recording the distances scaled up to the real court sizes.
- Practice reading off the distance from the central line where the ball hits the backboard, as this can be a little tricky.
- Place your toilet paper tube (player) at the 30° or 60° position. Put your ramp in place, put the ball on the ramp at the player position and let it roll. Note the point where the ball touches the backboard (wall). This is called the impact point. Read the distance from the central line to this impact point.
- Place the lower end of your ramp at different positions, leaving the position of the player (toilet paper tube) untouched. Roll your ball again and read the distance of the impact point to the central line.
- Make sure to note the location where the ball actually hits the backboard, and not the location where your cardboard track is pointing. These are not necessarily the same, as can be seen in Figure 10.
Close up photo of a small basketball bouncing off a wall that is marked with evenly spaced measurements. The impact point of the ball on the wall will need to be recorded so the measurement lines on the wall are drawn to a height above that of the basketball.
Figure 10. It is important to pay attention and make accurate readings of where the ball hits the backboard. In this picture, the blue arrow points to the impact point (close to 7 cm from the central line), while the cardboard track points to 3 cm, as indicated by the green arrow.
- Now you are ready to start testing. Start with the small distances from the central line (0, 1, 2, 3, and 4 cm). Start with the 90° position.
- Place the tube (player) on the player position.
- Note down in your lab notebook the position you are testing, such as "90° position, goal impact point at 0 cm" in a table like Table 3. Clear notes help you keep track of your results.
90° position, goal impact point at 0 cm. Trial Score Miss 1 2 3 4 ... 9 10 Totals:
Table 3. Table in which to record trials, scores, and misses for a particular player's position and goal impact point.
- As explained in step 2, aiming the track at a specific distance from the central line does not guarantee the ball will hit the backboard at that specific distance from the central line. In the following steps, you will use trial and error to aim the ramp and get the impact point you want to investigate.
- Aim the ramp as well as you can so the impact point will be at the distance from the central line you would like to test, starting with 0 cm from the central line.
- Let the ball roll and observe where the ball hits the backboard. If the impact position is not at the distance you want to investigate now, go back to step 3.d. and try again.
- Roll the ball and observe.
- If the ball did not roll smoothly or did not roll quickly enough to make a nice bounce back, disregard this trial.
- If the ball bounced back and rolled over the basket in a way that means it would have gone in, it is a score. Put a tally mark in the "Scores" column in the Trial 1 row of your table like Table 4. If it rolled away from or not directly over the basket, it is a miss. Put a tally mark next to "Misses" Note: You might think it would be easier to cut out the basket and watch if the ball enters the hole. This would be true if the ball did not roll over the basket on its way toward the backboard. You will cut out the basket later in this science project to test larger impact distances. You can retest these smaller distances if needed by putting the cut out circle back in place and placing the poster board back on the ground.
- Repeat step 3.f. until you have completed all ten trials.
- Add up the total number of "Scores" and note this result down in the correct spot in your table like Table 3. Copy your result in your data table like Table 2.
- Repeat steps 3.b.–3.h. for a distance of 1 cm to the central line, then again for a distance of 2, 3, and 4 cm to the central line.
- Repeat step 3, placing your player at each of the following positions: 60°, 30°, and 0°.
- Be sure you have the top four lines of your table like Table 2 filled in. Test any combination that is missing.
- Now that you have tested the impact distances close to the central line, the ball should no longer roll over the basket area on its way to the backboard, meaning you can now cut out the basket.
- Cut the basket out using a craft or utility knife. Adult supervision is recommended. Be sure to cut on a safe surface, like a cutting board, so you do not make cuts on anything like furniture or carpet underneath. Try to only cut on the outline of the circle (basket) so you can pop it back in and re-test the smaller impact distances if needed.
- Take two books of equal thickness (not too thick). Place them on either side of the cut-out basket, under the poster board. This gives the poster board a small elevation and allows the ball to fall into the basket. Note: Do not worry if the ends of the poster board flop down a little. It is important that the area of the poster board against the wall where the backboard is indicated (to the left of the basket) is lifted equally.
- Check if the central line lines up with the 0 cm line of the paper on the wall.
- Start testing the larger distances from the central line (5 cm and up). The procedure is very similar to step 3, except here, the ball will roll into the basket if it is a successful shot. Here is a quick recap of the steps. See step 3 for details.
- Put the player in position.
- Note the test position in your lab notebook by creating a table like Table 3.
- Direct the ramp at the desired test position and make test rolls until you get the desired impact distance.
- Perform a trial and place tallies to keep track of your results.
- Repeat step 7.d. until you have completed ten trials.
- Count the number of scores and write it down in your table like Table 3 and Table 2.
- Repeat steps 7.b.–7.f. for the other distances.
- Repeat step 7 for the following positions: 60°, 30°, and 0°.
Analyzing the Results
- Calculate the total number of successful banked shots for each angular position.
- Look back at your table like Table 2. Starting for the 90° position, add up the numbers in the column and write the result in the row "Total Number of Successful Banked Shots".
- Repeat step 1.a. for the 60°, 30°, and the 0° positions.
- Can you conclude from these results if a particular position(s) has a higher success rate?
- Are your results as you had expected? If you play basketball, do they confirm what you have experienced on the court?
- Make a scatter plot of the total number of successful banked shots (on the y-axis) with respect to the angular positions (on the x-axis).
- Calculate relative probabilities obtained from your scale model, using the 90° position as the baseline. Check the
Introduction if you need to refresh your memory on relative probabilities.
- Find the total number of successful banked shots for the 90° position in your data table. This is your baseline number.
- To obtain the relative probability, divide the total number of successful banked shots for a particular angular position by the baseline number. Note it down in your data table.
- Check if the relative probability for the 90° position is indeed 1, as this position is chosen as baseline. If not, find your mistake in step 1.a. or 1.b. and make corrections where necessary.
- Do your results show that it is easier (relative probability of higher than 1) or more difficult (relative probability lower than 1) to make a banked shot from specific angular positions with respect to the 90° position?
- Are your results as you had expected? If you play basketball, do they confirm what you have experienced on the court?
- Make a scatter plot of the relative probability (on the y-axis) with respect to the angular positions (on the x-axis).
- Analyze where on the backboard a ball must bounce to end up in the basket.
- Take a highlighter and highlight all the cells in your data table that have an entry of six or higher, as these combinations of angular position and distance to the center line resulted in a high rate of successful banked shots.
- Are these regions the same for all angular positions? Can you see a trend?
- Do some angular positions have a longer area on the backboard that leads to successful banked shots?
- Make a plot with the player's angular position on the x-axis and distance from the central line on the y-axis. For each angular position, draw a vertical line on the graph going from the minimum distance that scored at least six baskets, to the maximum distance that scored at least six baskets.
- For each angular position, calculate the width of the region where at least six baskets were scored. For example, if the closest distance to the central line where you scored six baskets was 5 cm, and the farthest distance was 12 cm, then the width is 12 − 5 = 7 cm. Now, make a new plot with angular position on the x-axis and this width on the y-axis.
- Relate back to the real basketball court by converting the distance of the impact point in your scale mode to a distance on a real court.
- Look back at section Create a Scale Model, step 1.d., for the conversion factor. This factor tells you how many centimeters in the scale model correspond to 1 m on the court.
- Use your scale factor to convert the distance from column "Ball Impact Position from the Central Line in a Scale Model" to the corresponding distance on the court. Hint: You need to divide by a scale factor to convert distances from the scale model to distances on the court.
- Note your results in the first column of your data table.
- Check if these distances seem plausible. Hint: Your results need to point to a location on the backboard, which is 72 inches or 1.83 m long.
- Repeat the scatter plot of step 3.d, this time indicating the distance on the real court on your y-axis.
- If you play basketball, do these distances correspond with your experience on the court?
- If you look up shot charts, model data, or other data indicating the success rate to make banked shots from different positions on the court, do your results confirm or contradict these results?
- If you doubt your data, you can increase your precision by adding more data points (for instance, add the 15°, 45°, and 75° data points) and/or make more trials for each combination (angular position, distance to the central line).
Ask an Expert
- In this science project, a distance of 3 m is used as distance to the center of the basket. Do the science project again, but this time, use a different distance to the basket. Note that you will need to recalculate to which distance the new chosen distance scales down, as well as adjust the player's positions on the poster board and the length of your track (cardboard tube).
- In this science project, you changed the position of the player by changing the angle and keeping the distance to the center of the basket constant. Do the project again, keeping the angle constant, but this time, analyzing the influence of changing distance to the basket.
- Recruit some of your basketball-playing friends and test their success rates from the analyzed positions, using the backboard. Compare the average of these real-life results with the results obtained in your scale model. Analyze what might account for the differences between your scale model and the real game.
- Look up "shot charts" for real basketball players (for example, do an online image search for "NBA shot chart") and compare your data to these shot charts. Does data from real basketball games match the trends you saw in your experiment, or not? What other factors do you think could influence a real game that were not present in your experiment?
- Use algebra and geometry to calculate the relative probability to score using the backboard from different positions on the court.
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