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Volume 7, Issue 1 2011 Article 3

Journal of Quantitative Analysis in

Sports

Optimal Targets for the Bank Shot in Men’s

Basketball

Larry M. Silverberg, North Carolina State University

Chau M. Tran, North Carolina State University

Taylor M. Adams, North Carolina State University

Recommended Citation:

Silverberg, Larry M.; Tran, Chau M.; and Adams, Taylor M. (2011) "Optimal Targets for the

Bank Shot in Men’s Basketball," Journal of Quantitative Analysis in Sports: Vol. 7 : Iss. 1,

Article 3.

Available at: http://www.bepress.com/jqas/vol7/iss1/3

DOI: 10.2202/1559-0410.1299

©2011 American Statistical Association. All rights reserved.

Optimal Targets for the Bank Shot in Men’s

Basketball

Larry M. Silverberg, Chau M. Tran, and Taylor M. Adams

Abstract

The purpose of this study was to gain an understanding of the bank shot and ultimately

determine the optimal target points on the backboard for the bank shot in men’s basketball. The

study used over one million three-dimensional simulations of basketball trajectories. Four launch

variables were studied: launch height, launch speed, launch angle, and aim angle. The shooter’s

statistical characteristics were prescribed to yield a 70 percent free throw when launching the ball

seven feet above the ground with 3 Hz of back spin. We found that the shooter can select a bank

shot over a direct shot with as much as a 20 percent advantage. The distribution over the court of

preferences of the bank shot over the direct shot was determined. It was also shown that there is an

aim line on the backboard independent of the shooter’s location on the court. We also found that at

3.326 inches behind the backboard, there exists a vertical axis that aids in finding the optimal

target point on the backboard. The optimal target point is the crossing of the vertical axis and the

aim line that is in the shooter’s line of sight.

KEYWORDS: basketball, bank shot, optimal, backboard

Introduction

For the spectator, the bank shot is distinctive and even a bit mystical. It demands

shooting a basketball farther than a direct shot and aiming the ball to the side.

Yet, most lay ups are bank shots and there are locations on the court where the

probability of a successful bank shot is considerably higher than the probability of

a successful direct shot.

Shooters perfect their bank shot technique by performing shooting drills.

Initially, however, the shooter can benefit from understanding the best launch

conditions. At what aim angle should the ball be launched? Where should the

ball make contact with the backboard? How does the contact point on the

backboard change with launch distance, launch angle, and launch height? These

are difficult questions to answer and, if left unanswered, prevent the shooter from

perfecting a most effective bank shot.

The optimal launch conditions for the bank shot are not obvious because

of the large number of factors. In practice, a prohibitively large number of bank

shots must be studied to gain a complete understanding of the optimal launch

conditions. An alternate approach is to perform computer simulations, where

millions of shots can be investigated in a relatively short amount of time.

Previous simulation studies of the basketball shot considered trajectories

launched from general locations on the court, as well as from the free throw line,

while apparently no detailed studies of the bank shot have been conducted. The

main contributors are Shibakuwa (1975), Brancazio (1981), Tan and Miller

(1981), Hamilton and Reinschmidt (1997), Huston and Grau (2003), and Tran and

Silverberg (2008). This paper studies the bank shot in detail and develops targets

on the backboard for the perfection of the bank shot technique.

Methods

Silverberg, Tran, and Adcock (2003) developed a general-purpose numerical

procedure for simulating basketball trajectories. Their model extended earlier

work as follows:

(1) The ball is assumed to be a thin lightly-damped elastic body that undergoes

rolling and/or sliding contact with the backboard and the rim.

(2) The ball undergoes any combination of consecutive bounces off the backboard

and the rim.

(3) The statistical characteristics of the skill level of the shooter are incorporated

in the procedure, making it possible to predict the probability of a successful

shot.

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Their model neglects three secondary effects. In order of decreasing

importance, the neglected effects are: vibration of the backboard and ring;

aerodynamic drag and Magnus force on the ball; and the bridge surface between

the backboard and ring. Their model has been tested extensively, producing

reliable results with errors in basketball simulations of less than 1%, and is used

throughout this paper. The dimensions of the court, backboard, and ring that

influence the bank shot are the same for international competition (International

Basketball Federation, 2006), US collegiate competition (National Collegiate

Athletic Association, 2001), and US professional competition (National

Basketball Association, 2006). However, the conclusions reached in the present

study apply only to men’s basketball because in woman’s basketball the ball is

smaller and lighter (Fig. 1).

Figure 1. Dimensions

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DOI: 10.2202/1559-0410.1299

In Figure 2, the ball is launched from a particular location that can be

expressed in terms of the rectangular coordinates (x y z) or equivalently in terms

of the cylindrical coordinates (r z) in which r denotes radial distance and

denotes polar angle. The coordinates are located at the center of the ball. The ball

is launched with a launch speed v, a launch angle and an aim angle . Notice

that is the angle between the plane of the trajectory and a horizontal line parallel

to the x axis. When = the player is shooting a direct shot The shooter also

imparts to the ball a back spin about an axis that is perpendicular to the vertical

plane of the ball’s approach to the basket. Out-of-plane components of back spin

can be imparted too, but these effects are neglected because of their typically

small magnitude.

Figure 2. Launch conditions

The shooter’s ultimate success depends on two factors. The first is his

understanding of the desired shot. Of course, the desired shot is not precisely the

optimal one. The second factor is the shooter’s consistency. The actual shot will

deviate from the desired shot because of the inevitable variability in shooting

movements. The selection of the desired shot and the standard deviations in the

launch conditions completely determine the chances of a shot being successful.

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Figure 3. Grid of 100 court locations

radial distance (ft): 1.969 3.281 4.593 5.905 7.218

8.530 9.842 11.15 12.47 13.75

polar angle (o): 0 10 20 30 40 50 60 70 80 90

In this study, bank shots are launched from the 100 court locations shown

in Fig. 3. From each location, a set of direct shots and a set of bank shots are

launched. The vertical planes of the trajectories are centered, that is, the vertical

planes pass through the center point of the ring. The aim angles of the centered

bank shots are determined from the formula shown in Fig. 4 (See Appendix). The

ball is launched 6 ft, 7 ft, and 8 ft above the ground. It was shown in the case of

the foul shot that imparting about 3 Hz (revolutions per second) of back spin is

optimal, so we let = 3 Hz here, too (Tran and Silverberg, 2008).

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

Figure 4. The aim angle of the bank shot

Figure 5 shows the launch speed v versus the launch angle for successful

bank shots from a set of 40,000 bank shots launched at r = 9.842 ft, = 60º and

7 ft above the ground. As shown, the region of successful shots has the shape of a

horseshoe. To gain a greater appreciation of the different types of bank shots

encountered, shots 1 through 11 are depicted around the figure. The trajectories

shown are the center-lines of the basketball. Shots 6, 5, 4, 1, 7, 8, and 9 are along

the outer (left) edge of the horseshoe, shots 10, 3 and 11 are along the inner (right)

edge, and shot 2 is in the middle of the horseshoe.

The shots along the outer edge have the smallest launch velocities for a

given launch angle and they first bank and then bounce off of the back of the ring.

The shots along the inner edge have the largest launch velocities for a given

launch angle and they bank and then bounce off of the front of the ring. Shot 2,

located in the middle of the v- region, is optimal (Tran and Silverberg, 2008). It

strikes the backboard and then swishes through the ring.

3

tanδtanβ

5γ

=

(a-R) tan

L

γ

x

v

3

5

y

v

a-R

backboard

R

Lcos+a-

R

)

tan

f

sinθ

tanβ3

cosθ15γ

a R

L

=

−

+ +

5

Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Figure 5. Launch speed versus launch angle for successful shots

Figure 6 shows the launch speed v versus the aim angle for successful

shots launched from another set of 40,000 bank shots launched again at

r = 9.842 ft, = 60º and 7 ft above the ground. The launch angle for all of these

shots is = 54º, which is the optimal launch angle located in the center of the v-

curve in Fig. 5. Note that the lines of constant probability in Figs. 5 and 6 (not

shown) are ellipses when v, , and are statistically independent and normally

distributed and when the other launch variables are regarded as deterministic.

The center of the largest probability ellipse fully contained in a region was taken

as the desired shot. The desired shot is the optimal shot when the probability of

that ellipse is sufficiently low (the probability of that ellipse is the calculation

error). The desired shots considered throughout the paper were all optimal.

(11)

(10)

(3)

(4)

(5) (6)

(7)

(8) (9)

(1) (2)

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

Figure 6. Launch speed versus aim angle for successful shots

Shot 1 has the lowest launch speed, shot 5 has the lowest aim angle, shot 3

has the highest aim angle, and shot 2, located in the middle of the v- region, is

optimal. Shots 1 and 4, which are launched at relatively low speeds, bounce low

off of the backboard. Shots 3, 5, 6, and 7, which are launched at relatively high

speeds, bounce high off of the backboard. Shot 2, the optimal shot that is

launched at a moderate speed, strikes the middle of the backboard. Also, the

optimal shot is centered; its aim angle produces a trajectory whose vertical plane

passes through the center point of the ring.

In the bank shot, the launch variable that the shooter finds particularly

difficult to select is the aim angle. To assist with aiming, the shooter benefits

from selecting a target point on the backboard toward which to aim. However,

when the polar angle is large the contact point C of the ball on the backboard is

(3)

(4)

(5)

(6)

(7)

(1) (2)

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

not aligned with the plane of the trajectory, making it difficult for the shooter to

aim toward (See Figure 7). To remedy this misalignment problem a shooter can

more naturally aim toward point A produced by extending the trajectory to the

backboard in the horizontal plane of the contact point. Point A is called the aim

point. Later in the paper, collections of aim points on the backboard, called aim

lines, will be studied.

Figure 7. The contact point C and the aim point A

Results

The nominal parameters are the 100 court locations, centered aim angles, optimal

launch angles, the launch height of 7 ft above the ground, and the back spin of

3 Hz. The results presented in this section use the nominal parameters and

deviations from the nominal parameters.

The Court

Figures 5 and 6 showed the v- curve and the v- curve for shots launched from a

single location on the court. The optimal (v was located in the middles of

the regions shown in these figures. We shall now assume that the optimal shot is

the shooter’s desired shot and that the shooter’s consistency, quantified in terms

of a standard deviation in launch speed (Tran and Silverberg, 2008), is a 70%

direct shot from the free throw line, which is about the average free throw

percentage in US collegiate competition as well as in the NBA. The same

standard deviation will be assumed for shots launched anywhere on the court,

backboard

C

A

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

although shooters actually tend to shoot more accurately from closer in. The 70%

shooting percentage is representative but arbitrary in that the same trends shown

below would be obtained for shooting percentages in the range of 65% to 85%

with the values scaled up or down. With these assumptions, the probabilities of

both the optimal bank shots and the optimal direct shots were calculated over the

indicated 100 court locations. The results are symmetric; differences between the

left and right sides of the court are being neglected. Note that the calculated

percentages tend to under-estimate the shooter’s performance since decreases in

the shooter’s standard deviations with distance to the ring are being neglected.

Figure 8a shows the probability of success of the bank shots. One

observes probabilities that decrease with distance, a peak occurs in the

neighborhood of polar angles of 75º where the backboard is fully utilized and a

second increase in the neighborhood of 0º where the bank shot and the direct shot

are aligned. As shown, the probability reaches over 90% close to the ring and

drops to 60% at 12 ft distances with angles in the neighborhood of 45º. Next,

referring to Fig. 8b showing the probability of success of the direct shots, one

observes probabilities that decrease with distance, and a peak occurs, again, in the

neighborhood of 0º. As shown, the probability of success at the free throw line is

70% (Recall, that is how the standard deviation in launch speed was set.), and

increases to more than 90% as the shooter moves closer to the ring. Finally, Fig.

8c shows the difference between the probability of success of the bank shots and

that of the direct shots. Therefore, a positive percentage indicates a level of

preference of the bank shot over the direct shot and a negative percentage

indicates the opposite. One observes that the bank shot is preferred in the red and

pink regions and the direct shot is preferred in the other regions. Notice that bank

shot preferences are on the order of 20% for polar angles of about 75º and in mid-

range distances for polar angles of 0º. The bank shot is not preferred at very short

distances to the ring where ball trajectories that bounce off of the backboard

require very large launch angles, nor preferred close to the foul line, nor at very

steep polar angles approaching 90º where the backboard is no longer effective.

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Figure 8: (a) bank shots, (b) direct shots, and (c) preferred shots

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

Note that these results do not take into account particular court conditions.

They mimic the conditions of a free throw shot during which play has been

stopped. The comparison between the bank shot and the direct shot discounts

such factors as the height and quickness of a defender, both of which influence

the player’s shot selection and hence change the presumed statistics.

Finally, keep in mind that the results presented above focus on

consistency, which is the second factor mentioned in the beginning of the paper

that determines whether or not a shot is successful. The first factor, the selection

of the desired shot, was treated by assuming that the shooter selects the optimal

shot as the desired shot. This is unattainable when the shooter does not have

knowledge of the optimal shot so, toward finding the optimal shot, the next sub-

section looks for the optimal targets on the backboard.

The Backboard

The locus of aim points on the backboard forms an aim line. The aim line is

associated with optimal shots launched from a given radial distance and a given

launch height from polar angles between 0º and 90º. Figure 9 shows aim points

(black) and corresponding contact points (green) for radial distances of 5.905 ft,

9.842 ft, and 13.75 ft. Note that the horizontal distance between a point on an aim

line and a point on a contact line increases with polar angle. At large polar

angles, the large distance corresponds to a misalignment of the ball’s trajectory,

illustrating the necessity for the aim line.

Also, note that the rectangle on the backboard provides some guidance as

to where the ball should make contact with the backboard. For an aim angle of

55º it was found that the contact point is close to the upper corner of the rectangle.

However, this does not imply that the shooter should aim toward the upper corner

of the rectangle because of the large misalignment between aim point and contact

point.

Furthermore Fig. 9 shows that the three aim lines corresponding to the

three radial distances are very close to each other. The vertical distances between

them (associated with one aim angle) is about ±2 inches. Although not shown, a

range of launch heights from 6 ft to 8 ft were also considered. Again, it was

found that these variations have little effect on the positions of the aim lines.

Indeed, the aim lines are approximately independent of the shooter’s location and

launch height. Therefore, there is practically a unique (averaged) aim line, as

shown in Fig. 10.

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Figure 9: Aim points (black) and contact points (green)

r = 13.75 ft (square), r = 9.842 ft (circle), and r = 5.905 ft (diamond)

Figure 10: Aim line

19.25 in

1.925 in

12.86 in

36.00 in

6.00 in

36.00 in

12

Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

As shown, the aim line is v-shaped with a flat bottom. At polar angles of

0º to 30º the shooter aims toward points on the flat bottom portion of the aim line.

At polar angles of 40º to 80º the shooter aims toward points on the v-shaped

portion of the aim line. This is an important result that helps the shooter

recognize where to target the ball. However, it stills remains to know at what

specific aim point on the aim line to direct the bank shot.

Training

Figure 10 demonstrated that there exists a single aim line on the backboard,

although no guidance was offered above as to the target point on that line. It turns

out, however, that the focal distance f in Fig. 4 is independent of the side angle ,

from which we conclude that the vertical planes of the optimal bank shot

trajectories all intersect at a single vertical axis f = 3.327 inches directly behind

the center of the backboard. These results suggest an approach toward training

players how to find the target point toward which to aim. The vertical axis could

be a physical pole behind the backboard and the aim line could be drawn on the

backboard, as shown in Fig. 11. The shooter could then look at the pole from any

court location and it will cross the aim line along his line of sight at the optimal

target point. Thus the shooter just aims toward the crossing.

Figure 11: Finding the targets using the pole and aim line

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

Conclusions

In this paper, bank shots launched from 100 court locations were studied. About

40,000 bank shots and another 40,000 direct shots were launched from each

location. These shots were launched from 7 ft above the ground, with 3 Hz of

back spin, and assumed a standard deviation in launch speed that corresponds to a

70% direct shot from the free throw line. Shots were also launched at other

launch heights to study the effect of launch height. In all, more than one million

shots were launched. Our results permit us to draw the following conclusions

about the bank shot in men’s basketball:

(1) A typical 70% free throw shooter can select a bank shot over a direct shot and

gain as much as a 20% advantage. This 20% advantage is significant in that a

70% shooter misses three times more than a 90% shooter. The court

preferences of the bank shot over the direct shot were given.

(2) The corner of the rectangle on the backboard corresponds to the optimal

contact point for an aim angle of 55º. The contact point is difficult to utilize

since it is not aligned with the direction of aim and applies to just one aim

angle.

(3) There exists a unique aim line on a backboard. The aim line is independent of

the shooter’s location on the court.

(4) The optimal target point can be pinpointed during a training session that

employs the pole and aim line. It is the crossing of the pole and the aim line

in the shooter’s line of sight.

The results presented in this paper can form the basis for future studies aimed at

establishing more effective ways of training players how to shoot the bank shot.

Appendix

The following shows that

which appeared in Fig. 4. Figure 12 shows the free body diagram of the ball

when it makes contact with the backboard. First refer to Fig. 12.

sinθ

tanβ3

cosθ15γ

a R

L

=

−

+ +

(1)

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

Figure 12: Free body diagram

The important equations are:

Equation (2a) follows from conservation of linear momentum in the x direction

and assumes a linear visco-elastic collision in which vx and vx’ denote the x

components of velocity just before and after contact, and is the coefficient of

restitution (Silverberg and Thrower, 2001). Equation (2b) follows from linear

impulse-momentum in the y direction in which m denotes ball mass, Fy denotes

the y component of force acting on the ball by the backboard, R is ball radius, and

vy and vy’ denote the y components of velocity just before and after contact.

Equation (2c) follows from angular impulse-momentum about the z axis in which

22

3

ImR= denotes the mass moment of inertia of a thin, spherical shell of radius

R and z’ denotes the angular velocity of the ball about the z axis just after

contact. Equation (2d) is a vector equation that expresses the kinematic

constraints between the velocity vector of the contact point just after the collision

backboard

C

x

y

Fy

Fx

'

x x

v

γv

=

(2a)

'

y y y

mv mv F dt

− =

∫

(2b)

ω'

z y

I R F dt

=−

∫

(2c)

/

' ' ' , that is

(0 0 0) ( ) ' (

ω ω ω ) ' ( 0 0)

C CG C CG

x y z x y z

v v v R

0 v v ωr

= = + ×

= + × −

(2d)

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

Published by Berkeley Electronic Press, 2011

vC’, which is zero, and the velocity vector of the center of the ball vCG’. By

expanding Eq. (2d), we get

'ω''ω'

yz z y

vR v R==-

(3a,b)

Substituting Eq. (2b) into Eq. (2c), and substituting the result into Eq. (3a), yields

3

'5

y

y

vv=

(4)

Next, referring to Fig. 4, we know that

tan β

y

x

v

v

= and 3

tan δtan β

5γ

= (5)

Also, notice that

Substituting Eq. (5) into Eq. (6), yields Eq. (1).

References

Brancazio, P. J. (1981). Physics of basketball. American Journal of Physics, 49,

356–365.

Hamilton, G. R. & Reinschmidt, C. (1997). Optimal trajectory for the basketball

free throw. Journal of Sports Sciences, 15, 491–504.

Huston, R. L. & Grau, C. A. (2003). Basketball shooting strategies – the free

throw, direct shot and layup, Sports Engineering, 6, 49–63.

International Basketball Federation (2006). Official basketball rules.

www.fiba.com.

National Basketball Association (2006). 2005–2006 Official rule book.

www.nba.com.

National Collegiate Athletic Association (2001). NCAA men's and women's

basketball rules and interpretations. www.ncaa.org.

Shibukawa, K. (1975). Velocity conditions of basketball shooting. Bulletin of the

Institute of Sport Science, 13, 59–64.

sin

θ= ( cos θ) tan β( ) tan δ

L L a R a R

+ − + −

(6)

16

Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 1, Art. 3

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DOI: 10.2202/1559-0410.1299

Silverberg, L. M. and Thrower, J. P. (2001). Mark’s Mechanics Problem-Solving

Companion, McGraw-Hill Book Company.

Silverberg, L. M., Tran, C. M., & Adcock, M. F. (2003). Numerical analysis of

the basketball shot. Journal of Dynamic Systems, Measurement, and

Control, 125, 531–540.

Tan, A. & Miller, G. (1981). Kinematics of the free throw in basketball. American

Journal of Physics, 49, 542–544.

Tran, C. M. and Silverberg, L. M., (2008). Optimal release conditions for the free

throw in men’s basketball. Journal of Sports Sciences, 26 (11), 1147 –

1155.

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Silverberg et al.: Optimal Targets for the Bank Shot in Men’s Basketball

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