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Mathematics in Sport

Maths helps athletes and organisers. A ball thrown at speed v and angle θ travels R = v² sin 2θ ÷ g (best at 45° without air drag). Statistics such as mean, median, batting average and strike rate summarise performance. Scheduling counts matches: a round-robin of n teams has n(n−1)/2 matches, a knockout has n−1.

🎬 Step-by-step story

  1. Throw a ball. Its path is a curve called a parabola. Here the angle is 40° and the speed is 14 m/s. Press Throw.
  2. Same speed, three angles: 30°, 45° and 60°. The farthest throw is at 45°. The throws at 30° and 60° land in the same spot.
  3. Data: a batter scored these runs in 7 matches. The orange line is the mean (the fair share).
  4. Change the last score. The mean jumps, but the median (the middle bar when sorted) hardly moves.
  5. Scheduling: every team must play every other team once. Slide the number of teams and count the matches.
  6. Free play: choose the angle and the speed, then throw. Try to land the ball on the red mark.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why is the path a curve and not a straight line?

Gravity pulls the ball down all the time while it moves forward at a steady speed. The mix of the two gives a parabola.

Why do 30° and 60° land at the same place?

sin 60° = sin 120°. Angles that add up to 90° have the same sin 2θ, so the same range. The grey and blue paths end at the same point in the 3D.

What does the mean line show?

If all bars were cut and shared equally, each would be the height of the orange line.

Why does the median hardly move?

The median only needs the middle bar. When the last score becomes very large, the order changes only a little near the middle.

Why do we divide by 2 in n(n − 1)/2?

Each team has n − 1 opponents, but a match between A and B is the same as B and A. Counting lines in the 3D, each pair has one line.

Can I hit the red mark exactly?

Yes. For a given speed there are two angles that reach the same distance. Try both, one low and flat, one high.

Sports mechanics: speed and projectiles

Speed = distance ÷ time. A sprinter who runs 100 m in 10 s has a speed of 10 m/s = 36 km/h (multiply m/s by 3.6 to get km/h).

A ball thrown, kicked or hit moves as a projectile. If we ignore air, gravity (g = 9.8 m/s² downwards) is the only force, and the path is a parabola. For launch speed v and angle θ above the ground, when the ball lands at the same height it started:

Since sin 2θ is largest (=1) when 2θ = 90°, the best angle is 45°. Angles that add up to 90°, like 30° and 60°, give the same range. In real sports, air drag and the release height change the best angle: a shot put is best near 40° because it is released above the ground. Read more in Projectile motion.

Data analysis in sport

To judge performance we summarise many numbers.

One very high or very low score (an outlier) pulls the mean a lot but hardly moves the median. So the median is safer when the data has outliers. Learn more in Mean, median and mode.

Scheduling tournaments

Organisers must decide how many matches and rounds are needed.

Try it: your own sports lab

Throw a soft ball at about 30°, 45° and 60° three times each and measure with a tape. Which angle goes farthest? Write the nine distances, find the mean and median for each angle. Then plan a league for the 6 teams of your class: how many matches, and how many rounds?

Key formulas and definitions

Worked examples

1. A sprinter runs 100 m in 12.5 s. Find the speed in m/s and km/h.

Speed = 100 ÷ 12.5 = 8 m/s. In km/h: 8 × 3.6 = 28.8 km/h.

2. A ball is thrown at 20 m/s at 30°. Find the range (g = 9.8 m/s²).

R = 20² × sin 60° ÷ 9.8 = 400 × 0.866 ÷ 9.8 = 35.3 m.

3. A ball is thrown at 14 m/s at 45°. Find the time of flight and the maximum height.

T = 2 × 14 × 0.7071 ÷ 9.8 = 2.02 s. H = 14² × 0.5 ÷ (2 × 9.8) = 98 ÷ 19.6 = 5 m.

4. A batter scored 400 runs in 10 innings and was not out twice. Find the batting average. Another innings: 60 runs off 40 balls. Find the strike rate.

Times out = 10 − 2 = 8, so average = 400 ÷ 8 = 50. Strike rate = 60 ÷ 40 × 100 = 150.

5. Scores: 24, 31, 18, 27, 35, 29, 22. Find the mean and median. Then replace 22 by 100 and find them again.

Sorted: 18, 22, 24, 27, 29, 31, 35. Mean = 186 ÷ 7 ≈ 26.6, median = 27. With 100: sorted 18, 24, 27, 29, 31, 35, 100. Mean = 264 ÷ 7 ≈ 37.7, median = 29. The mean rose by 11, the median by 2.

6. A league has 8 teams and each pair plays once. How many matches and rounds? How many matches if all play home and away?

Matches = 8 × 7 ÷ 2 = 28. Rounds = 7 (n even). Home and away: 8 × 7 = 56 matches.

7. A knockout has 12 teams. How many matches in total?

n − 1 = 11 matches. (Because 12 is not a power of 2, some teams get a bye in round one, and there are 4 rounds.)

Common mistakes

Practice quiz

1. Without air, which launch angle gives the longest range?
2. Matches in a round robin of 6 teams:
3. Which is less affected by one extreme score?
4. Strike rate of 30 runs off 20 balls:
5. Matches in a knockout with 16 teams:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

Why is 45° the best angle?

Range depends on sin 2θ. It is largest when 2θ = 90°, so θ = 45°. A throw at 30° and one at 60° land at the same spot.

Mean or median for a batter?

The mean gives the overall level. The median shows a typical score and is not pulled up by one huge innings. Using both gives the full picture.

Where is this topic taught?

It is an elective on maths in sport, for example in senior school maths in China (Mathematics in sport). Its ideas appear in physics (projectiles), statistics and discrete maths courses.

Where this is taught

South Korea고등학교 2학년Life and mathematics
China高三Elective D (PE and arts)

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