# 10 Olympic Starters

## Problem

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'?

## Student Solutions

Many of these problems involve an object moving under the influence of gravity and friction forces (such as air resistance). A first step in modelling is usually to neglect friction. In this case, the acceleration downwards due to gravity is about $9.8\mathrm{ms}^{-1}$. This leads to the simplifying modelling assumption that the downwards motion of a projectile is the reverse of the upwards motion. This can be used to calculate, for example, the speed of lift off by working out how fast an object will be landing when it hits the ground falling from a specified height (such as a pole-vault bar).

One of our summer students came up with these numbers:

1) Ignoring air resistance, if a long jumper could launch themselves at $45^{\circ}$ at a speed of $10.0\mathrm{ms}^{-1}$, they'd travel $\frac{10^2}{g}\mathrm{m}$ = $10.2\mathrm{m}$. (See here for a derivation). In reality, long jumpers would probably struggle to convert enough of their horizontal speed into the required vertical speed to travel on this trajectory. There's an interesting article here discussing how good a long jumper Usain Bolt could be.

2) Assume the gun is fired from exactly the same height as the target (though any height difference could be incorporated fairly simply into the model), and the bullet travels at a constant speed of $v= 400\mathrm{ms}^{-1}$. Suppose the bullet is fired at an angle $\alpha$ to the horizontal. The coordinates of the bullet are $$\left(v\cos(\alpha)t, v\sin(\alpha)t - \frac{g}{2}t^2\right)\;.$$ At time $t=T$ say, the bullet hits the target. If L is the distance to the target from the shooter, we find $\sin(2\alpha) = \frac{Lg}{v^2}$ (using $\sin(\alpha)=2\cos(\alpha)\sin(\alpha)$.) This gives the following values of $\alpha$ for the different distances:

L (m) |
$\mathbf{\alpha (^{\circ})}$ |

10 | 0.035 |

25 | 0.088 |

50 | 0.18 |

These seem to be very small numbers. However, if we included air resistance, the bullet would slow down during the motion and consequently hit the target lower than predicted here, so in reality there might need to be a noticably different aim for the different distances.

3) Suppose the athlete is only affected by the wind on the back and home straights, which are around $85\mathrm{m}$ in length, and they run exactly $1\mathrm{ms}^{-1}$ faster with the wind and $1\mathrm{ms}^{-1}$ slower when running against the wind. If the athelte's time in still conditions is $50\mathrm{s}$, then their average speed during the race is $8\mathrm{ms}^{-1}$. Their wind affected time using these assumptions would be:

$$\frac{(400 - 2\times85)\mathrm{m}}{8\mathrm{ms}^{-1}} + \frac{85}{8+1}\mathrm{s} + \frac{85}{8-1}\mathrm{s} = 50.3\mathrm{s}$$

i.e. their time is predicted to be $0.3\mathrm{s}$ slower.

We could incorporate other factors into the model. Air resistance, for example, is proportional to the speed squared, so could affect the time significantly. The wind would also affect the athlete around the bends of the track; here the wind strength would vary continuously. We've assumed the wind would affect the speeds along the straights symmetrically, in general this is likely to not be the case, as the athelete's performance may be reduced more than expected running into the wind.

4) A launch speed of $13\mathrm{ms}^{-1}$ seems a reasonable approximation. Neglecting air resistance, a throw with the velocity angled at $45^{\circ}$ to the ground would travel $\frac{13^2}{9.8}\mathrm{m} = 17.2\mathrm{m}$. The speed at which a shot putter can launch the shot depends on the launch angle. For more information, see here.

5) Suppose the athlete has just equalled the world record of $6.14\mathrm{m}$. Assume the athlete has zero vertical speed at the top of the jump. Their speed at the bottom of the descent is therefore $\sqrt{0^2+2\times9.8\times6.14}\mathrm{ms}^{-1} = 11.0\mathrm{ms}^{-1}.$

6) A footballer can kick the ball at a speed of around $75\mathrm{mph}$. Assuming the football remains at this speed, it'll take $11\mathrm{m}\times\frac{1}{75\times\frac{1600}{3600}\mathrm{ms}^{-1}} = 0.33\mathrm{s}$ to reach the goal. The goalkeeper needs to have moved into position and be ready to save it in this time!

7) Assume for the time being, the vertical velocity and trajectory result in the ball landing in the centre of the hoop looking from above. In the horizontal plane, the athlete can shoot at an angle $\alpha$ off the centre, where $\alpha = \tan^{-1}\left(\frac{0.23}{4.6}\right) = 2.8^{\circ}$. Can you work out any restrictions on the angle to the horizontal of the initial velocity?

8) Using $s=ut+\frac{1}{2}at^2$ shows that the time (for a point particle) to fall from $10\mathrm{m}$ is $\sqrt{2\times 10/9.8}\mathrm{s} = 1.43\mathrm{s}$. Thus, the time taken to travel to a height of $10\mathrm{m}$ and back is $2\times1.43\mathrm{s} = 2.86\mathrm{s}$. If they need to rotate $720^{\circ}$ in that time, they need to rotate at $252^{\circ}\mathrm{s}^{-1}$.

9) If we assume all his mass is $0.9\mathrm{m}$ from the bar, then conservation of energy gives: $mg(2\times0.9) = \frac{1}{2}mv^2$, so $v \approx 6\mathrm{ms}^{-1}$.

10) A 'singles' tennis court measures $8.3\mathrm{m}$ by $23.8\mathrm{m}$. Suppose the player can serve at $50\mathrm{ms}^{-1}$. The straight serve takes $\frac{23.8}{50}\mathrm{s} = 0.476\mathrm{s}$ to get to the baseline. The diagonal serve travels $\sqrt{4.15^2 + 23.8^2}\mathrm{m} = 24.2\mathrm{m}$, so takes $0.49\mathrm{s}$.

## Teachers' Resources

### Why do this problem ?

Mathematical modelling is an important skill and mechanics courses are often the first places in which modelling can be explored. These interesting questions will allow you to practice these skills whilst developing awareness of the key concepts of mechanics.

### Possible approach

There are several parts to this question. The individual pieces could be used as starters or filler activities for students who finish classwork early. Enthusiastic students might work through them in their own time. Since there is no absolutely 'correct' answer to many of these questions, they might productively be used for discussion: students create their own answers and then explain them to the rest of the class. Does the class agree? Disagree? Is there an obvious best 'collective' answer?

### Key questions

- What assumptions will you need to make in this question?

- What equations of mechanics will you need?

- How accurate do you think you answer is?

- What order of magnitude checks could you make to test that your answer is sensible?

### Possible extension

Can students make up similar questions? Can they put any upper or lower bounds on the numbers?

### Possible support

Students might struggle with the 'open' nature of the questions. To begin, they might like to read the Student Guide to Getting Started with rich tasks