Sports & Motion Analytics

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

Sports & Motion Analytics brings together the formulas that turn movement, performance, and probability into numbers you can measure and compare. From projectile range and hang time to pace, xG, Elo ratings, and cricket or baseball stats, these equations help students, exam candidates, coaches, analysts, and curious fans make sense of real-world sport. They matter because they turn instinct into insight and performance into data you can trust.

Projectile Range

NEWIntermediate

R = \frac{v^2 \sin(2\theta)}{g}

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

Projectile Max Height

NEWIntermediate

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

Peak height reached by a launched projectile.

Hang Time

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

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

Running Pace

P = \frac{t}{d}

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

Speed from Pace

v = \frac{60}{P}

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

Expected Goals (xG)

NEWIntermediate

xG = \sum_{i} p_i

Sum of scoring probabilities of every chance a team creates.

Elo Rating Update

NEWIntermediate

R' = R + K(S - E)

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

Elo Expected Score

NEWIntermediate

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

Probability player A beats player B given their ratings.

Batting Average (Cricket)

Avg = \frac{R}{D}

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

Strike Rate (Cricket)

SR = \frac{R}{B} \times 100

Runs scored per 100 balls faced — measures scoring speed.

Bowling Economy

Econ = \frac{R}{O}

Runs conceded per over bowled.

Baseball Batting Average

BA = \frac{H}{AB}

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

VO₂ Max (Cooper Test)

NEWIntermediate

VO_2 = \frac{d - 504.9}{44.73}

Estimates aerobic fitness from distance run in 12 minutes.

Power-to-Weight Ratio

NEWIntermediate

PWR = \frac{P}{m}

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

Win Percentage

W\% = \frac{W}{G} \times 100

Fraction of games a team has won.

Pythagorean Expectation

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

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

Points Per Game

PPG = \frac{P}{G}

Average points a player or team scores per game.

Golf Handicap Differential

D = (S - CR) \times \frac{113}{SR}

Standardises a round score against course difficulty.

Reaction Distance

d = v \times t_r

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