Levers & reaction
Torque = force × distance from the pivot (N·m). A lever balances when F₁ × d₁ = F₂ × d₂. Time = distance ÷ speed; 1 s = 1,000 ms.
Ruler drop: d = 490.5t² (d in cm, t in s), so t = √(d ÷ 490.5). A stance tips past tan θ = half the base ÷ height of the centre of mass. Try the ruler drop only with a partner, sitting down.
One step
Torque, timing and scores.Torque
A force of 80 N acts 0.25 m from a pivot. What is the torque?
Fast hands
A hand moves 0.6 m at 10 m/s. How long does that take, in milliseconds?
Boxing card
A judge scores three rounds 10–9, 10–9 and 9–10. What is the final card?
Taekwondo round
Athlete A: turning head kick, punch, body kick. Athlete B: head kick and body kick, and A got one gam-jeom. Use the p.14 table. What is the score?
Two or three steps
Sketch it and label the units.Stable stance
Find the tipping angle tan⁻¹(half-base ÷ height) for a half-base of 0.25 m with the centre of mass 0.90 m up. Then for 0.35 m and 0.75 m. Which is more stable?
Ruler drop
A partner catches a dropped ruler at 12 cm. What was their reaction time? What if it falls 22 cm?
Reach
An arm reaches 0.60 m from the shoulder, and a step covers 1.60 m. How far away can a touch land?
Weight classes
Women’s Olympic judo classes start 48, 52, 57 and 63 kg. Find the three percentage gaps. What pattern do you see?
Belt schedule
The p.12 example takes 48 months to black belt. A school takes 25% longer at every step. How long is that, in months and years?
Judo score
Blue has 1 waza-ari; White has 2 yuko. Who leads? White then scores a waza-ari. Who leads now, and why?
Equations
A1–A2 (pp.17–18).Balance a lever
A 240 N force acts 0.15 m from a pivot. What force at 0.45 m balances it? Solve 240 × 0.15 = F × 0.45.
Find the arm
A 200 N force needs to make 36 N·m of torque. Solve 200d = 36.
Reaction time, squared
Someone catches a ruler at 24.5 cm. Solve 24.5 = 490.5t² for their reaction time in ms.
Long enough?
To test reaction times up to 0.35 s, how long must the ruler be? Use d = 490.5t².
Your turn to be the sports scientist
- Pick a combat sport or martial art you know about.
- Write a question about its scores, angles, timing or weight classes.
- Solve it, and say what data would make your answer fairer.
My problem, and my worked answer:
Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.
The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.
Answer key
Worked solutions. Practise any skill only with a qualified instructor; this sheet is about the maths, not technique.