Math and sports

Think of any ways math can be used in basketball?Think of any ways math can be used in basketball?... Math in BasketballPlayer must throw the ball about 19 feet per second at a 45 degree

Math and Sports

Paul Moore April 15, 2010

Fantasy Sports

Playing Sports

Real World Applications

Velocity & angle of shots

Physics equations and derivation

Think of any ways math can be used in basketball?

Think of any ways math can be used in basketball?

t = time

a = accelerationThis is 490….where did this equation come from?

Math in Basketball

By definition: Average velocity

vavg = Δs / t

= (s – s0) / t Assuming constant acceleration

Assuming constant acceleration

vavg = (v + v0) / 2 ) / 2

Combine the two:

(s – s0) / t = (v + v0) / 2

Δs = ½ (v + v0) t

v = v0 + atSubstitute velocity into Δs equation above

sy

sx

Math in Basketball

(sx): x = x0 + v0xt + ½axt2(sy): y = y0 + v0yt + ½ayt2

Need further manipulation for use in our real world application

Often will not know the time (like in our example here) or some other variableHere:

– ax = 0, x0 = 0

– ay = -32 ft/sec2

(sx): x = v0xt (sy): y = y0 + v0yt + (-16)t2

Math in Basketball

(sx): x = v0xt (sy): y = y0 + v0yt + (-16)t2

Next, want component velocity in terms of total velocity

(sx): x = v0 cosθt (sy): y = y0 + v0sinθ t + (-16)t2

Math in Basketball

(sx): x = v0 cosθt (sy): y = y0 + v0sinθ t + (-16)t2

Don’t know time…

Solve x equation for t and plug into y

t = x / (v0 cosθ )

…into y equation…

y = y0 + v0sinθ [ x / (v0 cosθ ) ] + (-16)[ x / (v0 cosθ ) ]2

y = y0 + x tanθ + (-16)[ x2 / (v02 cos2θ ) ]

We know initial y, initial x, final x, and our angle

Now we have a usable equation!

Math in Basketball

y = y0 + x tanθ + (-16)[ x2 / (v02 cos2θ ) ]

Distance: x = 14 ft Initial height: y0 = 7 ft (where ball released)

Final height: y = 10 ft Angle: θ = 45 Find required velocity: v0

7 = 10 + (14)tan(45) – 16[ 142 / (v02cos2(45)) ]

7 = 10 + 14 – 3136 / (0.5 v02)

17 = 6272 / v02V0 = 19.21 ft / sec

Math in Basketball

Player must throw the ball about 19 feet per second at a 45 degree angle to reach the basket

This, of course, wouldn’t guarantee the shot will be made

There are other factors to consider:

– Air resistance

– Bounce of the ball on the side of the rim

Math in Baseball

“Sweet Spot” of hitting a baseball

– Frequency and

intensity depend on location of contact

– Vibration is really

energy being transferred from ball to the bat (useless)

Math in Baseball

Sweet spot on bat where, when ball contacts, produces least amount of vibration…

– Least amount of energy lost, maximizing energy

transferred to ball

Math in Baseball

Pitching a Curve Ball

– Ball thrown with a downward

spin Drops as it approaches

plate

For years, debated whether

curve balls actually curved

or it was an optical illusion

With today’s technology,

it’s easy to see that they

do indeed curve

Math in Baseball

Curve Ball

through the air

one side of the ball

Math in Baseball

Batting

90 mph fastball takes 0.40 seconds to get from the pitcher to the batter

If a batter overestimates by 0.013 second swing will be early and will miss or foul ball

What’s the best speed/angle to hit a ball?

Math in Baseball

Use the same equations:

(sx): x = x0 + v0xt + ½at2(sy): y = y0 + v0yt + ½at2

Use the same manipulation to get:

y = y0 + x tanθ + (-16)[ x2 / (v02 cos2θ ) ]

Let’s compare velocity (v0) and angle (θ)

…solve for v0

Math in Baseball

y = y0 + x tanθ + (-16)[ x2 / (v02 cos2θ ) ]

Solved for v0 (ft/sec)

At a particular ballpark, home run distance is constant

– So distance (x) and height (y) are known

θ

0

2 0

cos )

tan (

16

x y

y

x v

+

−

=

Math in Baseball

Graphing solved function with known x and y compares velocity with angle of hit

– shows a parabolic function with a minimum at 45

Math in Baseball

) 35 ( cos ))

35 tan(

) 500 (

20 3

(

) 500 (

16 cos

) tan (

16

2

2

20

20

+

−

= +

−

=

θ θ

x y

y

x v

775

133 812

.

17895 516

223

Math in Baseball

Previous examples do not incorporate drag or lift

Graphs with equations including drag and lift:

Optimal realistic angle:

about 35 degrees

Stats in Baseball

Baseball produces and uses more statistics than any other sport

Evaluating Team’s Performance

Evaluating Player’s Performance

Coaches and fantasy players use these stats to make choices about their team

2010 Season Stats

SPLITS G AB R H 2B 3B HR RBI BB SO SB CS AVG OBP SLG OPS Season 8 23 8 8 2 0 2 5 8 2 1 0 348 516 696 1.212 Career 619 2146 271 631 158 2 93 394 208 303 13 3 ? 358 500 858 Last 7 days 6 18 4 6 2 0 1 4 4 2 0 0 333 455 611 1.066 Projected 162 466 162 162 41 0 41 101 162 41 20 0 348 516 696 1.212

Stats in Baseball

Some Important Stats:

Batters

– Batting Average (BA)

– Runs Batted In (RBI)

– Strike Outs (SO)

– Home Runs (HR)

Pitchers

– Earned Run Average (ERA)

– Hits Allowed (per 9 innings) (H/9)

– Strikeouts (K)

Stats in Baseball

Batting Average (BA)

BA ( ) =

Stats in Baseball

Runs Batted In (RBI)

Earned Run Average (ERA)

nine innings

Hits Allowed (H/9)

nine inning period

pitched innings

allowed

hits H

_

) 9 _

( 9

Pitched Innings

Allowed Runs

Earned ERA

“Soccer is a game of angles”

Goaltending vs

Shooting

Where should goalie stand to best defend a shot?

Angles in Soccer

May think it best to stand in a position that bisects goal line

Gives shooter more room between goalie and left post, than right post

Angles in Soccer

As distance from goal increases, the angle bisection approaches the goal line bisection

Sports & Math Education

Incorporation and application of math in sports is a creative, and wildly successful method of teaching mathematics

Professors, University of Mississippi taught fantasy football to 80 student athletes Before, 38% received A’s on a pretest After, 83% received A’s on a postest

http://www.fantasysportsmath.com/

Sports & Math Education

Innovative way to get students doing math

Even if some are not interested, they’re able to understand the practicality and application of mathematical concepts

What sports did you all play?

Can you think of any other ways math is involved in sports?

Do you think incorporating sports is an effective method of teaching mathematics?