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Sports & Motion Analytics

The maths behind the game: projectile range, pace, xG, VO₂ max, Elo and player ratings

Projectile Range

NEWIntermediate
R=v2sin(2θ)gR = \frac{v^2 \sin(2\theta)}{g}

Horizontal distance a ball travels when launched at speed v and angle θ.

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Projectile Max Height

NEWIntermediate
H=v2sin2θ2gH = \frac{v^2 \sin^2\theta}{2g}

Peak height reached by a launched projectile.

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Hang Time

NEWBasic
t=2vsinθgt = \frac{2v\sin\theta}{g}

Total time a projectile spends in the air on level ground.

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Running Pace

NEWBasic
P=tdP = \frac{t}{d}

Time taken per unit distance — the runner's pace.

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Speed from Pace

NEWBasic
v=60Pv = \frac{60}{P}

Convert a running pace in min/km into speed in km/h.

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Expected Goals (xG)

NEWIntermediate
xG=ipixG = \sum_{i} p_i

Sum of scoring probabilities of every chance a team creates.

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Elo Rating Update

NEWIntermediate
R=R+K(SE)R' = R + K(S - E)

Updates a player's rating after a game based on result versus expectation.

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Elo Expected Score

NEWIntermediate
E=11+10(RbRa)/400E = \frac{1}{1 + 10^{(R_b - R_a)/400}}

Probability player A beats player B given their ratings.

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Batting Average (Cricket)

NEWBasic
Avg=RDAvg = \frac{R}{D}

Runs scored per dismissal — a batter's core stat.

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Strike Rate (Cricket)

NEWBasic
SR=RB×100SR = \frac{R}{B} \times 100

Runs scored per 100 balls faced — measures scoring speed.

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Bowling Economy

NEWBasic
Econ=ROEcon = \frac{R}{O}

Runs conceded per over bowled.

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Baseball Batting Average

NEWBasic
BA=HABBA = \frac{H}{AB}

Hits divided by at-bats — baseball's classic hitting measure.

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VO₂ Max (Cooper Test)

NEWIntermediate
VO2=d504.944.73VO_2 = \frac{d - 504.9}{44.73}

Estimates aerobic fitness from distance run in 12 minutes.

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Power-to-Weight Ratio

NEWIntermediate
PWR=PmPWR = \frac{P}{m}

Cyclist's sustainable power divided by body mass — key to climbing.

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Win Percentage

NEWBasic
W%=WG×100W\% = \frac{W}{G} \times 100

Fraction of games a team has won.

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Pythagorean Expectation

NEWAdvanced
W%=PF2PF2+PA2W\% = \frac{PF^2}{PF^2 + PA^2}

Predicts a team's win rate from points scored and allowed.

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Points Per Game

NEWBasic
PPG=PGPPG = \frac{P}{G}

Average points a player or team scores per game.

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Golf Handicap Differential

NEWAdvanced
D=(SCR)×113SRD = (S - CR) \times \frac{113}{SR}

Standardises a round score against course difficulty.

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Reaction Distance

NEWBasic
d=v×trd = v \times t_r

How far you travel during reaction time before braking — vital in motorsport and driving.

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