The Physics of Basketball

Ulrike LahaiseGeorgia Perimeter College

Decatur Campus Symposium 2006

Outline• Little Tidbits• Intro

– Projectile Motion– Concepts of Work and

Energy

• The Swish– Lay Up Shot– Best Launching Angle– Wilt’s Problem/Solution– Observation of Real Play

• The Spin• The Bounce• Bibliography

Little Tidbits I

Why was Wilt Chamberlain bad at foul shots?

• 7 ft. 2 in.; Philly Warriors; ‘59/60 – ‘72/73; 31,419 career points; 23,924 reb. in 1045 games; lifetime avg. 30.1 pts./gm.; 22 reb./gm; 45.8 min/game in playing time.

• poor foul-shooting ability, 51.1%! • Hypothesis: because he was so tall?

– Angelo Armenti (“The Physics of Sports”)

• Observation: tall players have worse foul shot percentage than short players

• Not as bad as Wilt’s, though!

Little Tidbits I

From “The Physics of Sports”, edited by Angelo Armenti

Little Tidbits II1980’s Nike Commercial• Mars Blackman: Yo Professor, how does

Mike defy gravity, do you know, do you know, do you know, do you know?

• Prof: Michael Jordan overcomes the acceleration of gravity by the application of his muscle power in the vertical plane thus producing a low altitude earth orbit.

• Mars: A what?• Prof: Do you know what I mean? Do you

know, do you know, do you know?• Mars: Money, check him out!

Little Tidbits II

From “The Physics of Sports”, edited by Angelo Armenti

Little Tidbits II

Hang Time - Translation• Jordan “flies” in two dimensions:

horizontally and vertically• less height/distance than the world

records – high jump (around 2 m): only vertical– long jump (around 9 m): only horizontal

• full force at angle: vertical and horizontal components combined– Equally impressive!

Intro: Projectile Motion (I)

• Two dimensional motion of an object released with a certain launching speed/angle

• Neglect air resistance• Only acceleration of object is Earth’s

gravity• Split up into horizontal motion component

and vertical motion component• Dr. John Evan’s slides

Intro: Projectile Motion (II)

• Horizontal motion is constant• Vertical motion is uniformly accelerated• Vertical position vs. time is a parabola• Vertical position vs. horizontal position

(shape of the path) is a parabola, (figure from Peter Brancazio)

Intro: Projectile Motion (III)Demos and Visuals• Cutnell & Johnson College Physics• Projectile Motion Simulation

(http://www.hazelwood.k12.mo.us/~grichert)

• R. Nave (HyperPhysics – Mechanics)

Intro: Work of a Shot (I)

How does the player impart speed?• Player’s work: work = force x

distance in hand• Player’s work = change in ball’s KE

– KE = Kinetic Energy = energy of motion• Depends on mass and square of speed

– Can get same increase in KE by varying force or distance

Intro: Work of a Shot (II)

Example: Granny (or, in this case, “Daisy”) Shot

• Underhanded shooting style– If force is lacking make it up with distance!

The Swish: Lay Up Shot (I)

• Ball shot on the run– maintains same

speed and direction as player had when shooting (due to its inertia)

– Plus, speed and direction given by player’s shot

– Image from Worsley School, Alberta, Canada, Science Files

The Swish: Lay Up Shot (II)

Ex.: player shoots while running to net

• push ball toward basket?– Ball has same forward motion as player,

likely overshoots with additional forward push

• shoot it straight up? – shoot straight up when 2 – 3 feet away from

net, score if launching speed and timing are right

The Swish: Lay Up Shot (III)

Ex.: player shoots while running crosswise

• Aim at center of basket?– Ball overshoots, bounces off far edge of

rim

• Aim at near side of rim? – Compensates for ball’s inertia in same

direction as players motion; best chance of going in

The Swish: Free Throw

Problem: Best Launching Angle (Peter Brancazio)

• given distance: infinite # of speed/angle combinations to center of baskets

• shape of path (“trajectory”): parabola• One optimum speed/angle combination

– most leeway (wiggle room) – Lowest amount of force needed

The Swish: Theory (I)

• entry angle between direction of ball and plane of rim ≥ 32°– Angle = 32°

• No leeway, area ball “sees” = its own

– For angles between 32° and 90°• Leeway increases,

area ball “sees” > its own

The Swish: Theory (II)

• Minimum launching angle for minimum entry angle– Need specific launching speed to make the shot– Is smaller for taller players and larger distances

The Swish: Theory (III)• For launching angle greater than minimum

– More leeway in launching speed: ball can make basket within range of launching speeds

– Left/right leeway is greater for shorter distances and the higher release height

The Swish: Theory (IV)Launching angle hard to measure, instead:• distance, L: 10 – 25 ft.

– Any closer, player might bank; further shots - desperation

• release height below basket, h: 1 – 4 ft.– Players shoot 1 – 2 ft. above head including

jump shots

The Swish: Theory (V)

• Scenario: 6 ft player; L = of 13.5 ft.; h = 2 ft• Leeway in launching speed (v) very small!

– Doesn’t get larger than ~ 1%– Slightly increases with increasing launching angle

but not significantly: player must be accurate

• Leeway in launching angle greater than for v– largest for angles right around minimum speed

angle (49.2°)– Between 4 and 11%

• Larger shooting distances (L): smaller leeway

The Swish: Wilt’s Problem

• Same L and shooting height closer to net (h): larger leeway – Theoretically: Taller players have a

better chance – Wilt’s problem according to Peter

Brancazio: “Fortunately for the shorter players, most taller players tend to develop their rebounding and under-the-basket play at the expense of their longer-distance shooting skills.”

The Swish: Wilt’s Solution (I)Overhand vs. Underhand Shot• Theoretical calculations by A. Tan

and G. Miller for avg. NBA player, 6 ft 6 in

The Swish: Wilt’s Solutions (II)• Overhand shot:

– Lower launching speed: less force required (+)

– Larger entry angle: ball “sees” larger area when entering basket (+)

– Larger entry speed: higher chance to bounce off rim if clean shot missed (-)

• Underhand shot:– Lower entry speed: less chance to bounce

off rim if clean shot missed (+)– Higher launching speed: more force

required (-)– Smaller entry angle: ball “sees” smaller

area to enter (-)• Can increase launching angle/speed with more

force

The Swish: Wilt’s Solution (II)

• Theoretical Prediction: Overhand shot preferable

• Experiment: Underhand shot more successful – Ball is evenly balanced between the two

hands – Muscles are more relaxed – Greater accuracy in the forward direction:

entire arm goes into the swing, less player error when aiming

– Types of muscles in overhand shot harder to control: shot directed with one hand less accurate in forward direction

The Swish: Wilt’s Solution (III)

Observation: Attitude triumphs!• Underhand shot

– Advocated by former NBA player Rick Barry– Wilt used it for a while and improved his

stats– Minimizes left/right drift of ball– Player has more control over shot

• Overhand Shot– requires movement from the wrist, elbow,

and shoulder: More ways to mess up!– Overriding plus: It looks a lot cooler!

The Swish: Observation (I)• Players develop “kinesthetic”, aka muscle

memory– Able to make tiny adjustments to correct shot– average pro hits ~ 50% under game

conditions, > 70% when unguarded, best pros ~ 90% when unguarded

• Different launching angles for same speed – 60° yields high arch, 30°yields low (or flat)

arch– 45° yields furthest distance

The Swish: Observation (II)Launching angle: high arch vs. low arch• High arch angle > minimum speed angle

– Slightly larger leeway in speed, harder to block by tall defenders (+)

– Harder to launch and aim (-)

• Low arch angle < minimum speed angle– youngsters on playground lacking height and

strength – Shot aimed directly at rim– very small leeway in launching speed (-) – larger launching speed than minimum-speed

angle (-)

The Swish: Observation (III)

• Launching angle used by pros:– Early days: slower,

more deliberate play; set shot most often used

– Set shot: a high-arch shot launched from the chest or over the head with both feet on the floor

The Swish: Observation (IV)• Today: game is faster and more “one-on-one”• Jump Shot most commonly used

– launched while in mid air usually with one hand– launched quickly with minimum effort – used even when unguarded rather than set shot– Theory: more leeway (launched higher up)– In reality: players launch it as “low-arch” shot!

• But slightly higher arch: higher accuracy and less force

– Players use same launching angle regardless of L• Theory: minimum-speed angle larger for smaller L• higher-arch shot nearer net gives more leeway

The Swish: Animation

• Make a jump shot every time:– “Fear of Physics” web site– http://www.fearofphysics.com/Proj/

proj.html

• Using basketball to teach physics and geometry concepts– Basketball Explorations: Interactive web

site developed by a highschool group– http://library.advanced.org/12006/

The Spin (I)

• Ball shot with fingertips (and flick of wrist)– Better control of launching angle– Ball gets backspin

• Arnold “Red” Auerbach – great basketball coach – “backspin makes the shot softer and

helps it to be “lucky””• Peter J. Brancazio: “Good physics

rather than good luck!”

The Spin (II)• Ball strikes surface/rim with specific angle and

speed• Can split balls motion into horizontal and

vertical components• Certain energy in both types of motion

– Add up to total energy of motion– Bounce: spin shifts some energy of one type to other– Some energy will be lost (shifted to the surface)

The Spin: No Initial Spin

• frictional force opposes the forward, horizontal motion

• acts as a torque: ball gets forward spin– forward energy less, spin energy more – rebound with forward spin, not quite as

fast in the horizontal direction

The Spin: Initial Top Spin• if ball spins fast enough, backward

motion of ball’s bottom faster than its forward motion– frictional force opposes the net backward

motion of the ball’s skin, acts forward – Some energy shifted from spin to forward

motion– rebound with less forward spin, more

forward motion: faster, lower angle

The Spin: Initial Back Spin

• ball as whole has forward motion and its bottom has forward motion due to spin– Frictional force opposes both kinds of

motion– spinning and forward motion decrease– bounce straighter upward, larger angle– considerable speed loss– More likely to score

after hitting rim or backb.

The Bounce (I)• Deformation of surface

of ball depends on air pressure inside– The more air pressure,

the less deformation• The more energy stored

in compressed air inside

– Air stores and returns energy more efficiently than material of ball

The Bounce (II)• Under inflated ball: some energy wasted

in deformation – Ball material heats up due to internal friction

• Molecules move across one another – ball heats up

– ball will not rebound very high

• Highly pressurized ball: more elastic collision– Floor must be harder than ball!– soft floor material will flex when the ball hits

it, ball looses some energy to floor• Floor heats up

The Bounce (III)

Example: 4th grade science fair project• Bounce height of balls measured as function of

internal air pressure: 0 psi, 3 psi, 6 psi, 9 psi, 12 psi

• Bounce height increased with pressure, highest bounce for 9 psi, a little lower for 12 psi

• U of Virginia explanation: reasonably hard surface used, but it began to dent significantly for 12 psi-ball– 12 psi-ball hard enough to dent floor– Floor took part in bounce, ball shifted some of its KE

to it

Do’s and Don’t’s

• Do attempt to score

• Don’t travel with ball!

Bibliography• “The Physics of Sports”, collection of articles edited

by Angelo Armenti Jr.– “The Physics of Basketball” by Peter J. Brancazio– “Kinematics of the free throw in basketball” by A. Tan and

G. Miller• John Evan’s Introductory Physics Power Point

Slides: http://www.complexsystems.us/• Cutnell & Johnson “College Physics”:

http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471151831/concepts/index.htm?newwindow=true

• Projectile Motion Simulation: http://www.hazelwood.k12.mo.us/~grichert

• Rod Nave, Georgia State, HyperPhysics – Mechanics: http://hyperphysics.phy-astr.gsu.edu/Hbase/hph.html

• Worsley School, Alberta, Canada, Science Files: http://www.worsleyschool.net/science/files/physicsof/basketball.html

• University of Virginia: http://howthingswork.virginia.edu/bouncing_balls.html