These 10 questions encourage students to explore mathematical modelling in the context of several different sports including track and field athletics, shooting, football, tennis, basketball and gymnastics. This advanced mathematics activity is designed to be accessible to A-level mechanics students (Key Stage 5).

Detailed teachers' notes for this activity are available on our NRICH website.

##### Stage: 5 Challenge Level:

Consider some of these questions concerning the mechanics of sport. You might need more data in some cases or need to make an approximation to allow for mathematical modelling. You might be able to give precise answers or answers bounded by some reasonable error range. Be as precise as you can in your assumptions so as to convince yourself or others of the answers.

1. What if a long jumper could launch him or her self from the platform at 45 degrees with the same speed as at their standard launch angle? How far would they jump?

2. In pistol and rifle events, competitors aim at a 10-ringed target from the set distances of 10m, 25m and 50m. Do you think that marksmen need to alter their angle of aim by a measurable amount between these targets?

3. Imagine that a wind of speed 1ms$^{-1}$ is blowing parallel to the straight parts of the athletics track. Do you think that this would help or hinder a 400m sprinter?

4. What if a shot-putter could launch the shot at an angle of 45 degrees at the same speed as their usual launch angle?

5. At what speed does a pole-vaulter hit the crash mat?

6. In football, a penalty is taken 12 yards away from the goal. How good do the goalkeeper's reactions have to be?

7. A basketball free throw is taken 4.6m from the hoop. The hoop is 45.7cm in diameter, and 3.05m high. The basketball is 24cm in diameter. How precise does a player's shot have to be to ensure the ball goes in the hoop?

8. A trampolinist can jump to a height of 10m. They perform a double somersault. How quickly must they be able to rotate in order to land safely on their feet and not on their head?

9. A gymnast is swinging on a high bar. The distance between his waist and the bar is 0.90m. At the top of the swing his speed is momentarily 0ms$^{-1}$. Calculate his speed at the bottom of the swing.

10. Assuming the ball travels at a constant speed throughout, how much longer does a tennis serve to the edge of the court take to reach the baseline than a serve 'down the T'?

**Further information**

Detailed teachers' notes for this activity, including sample solutions, are available on our NRICH website.