Upper Middle School - athletics (Mech)


1. Two staff, ABC and XYZ are struggling with the unfamiliar task of taking the times on Sports Day. They decide that ABC will take the first two positions and XYZ the other two. In one of the 100m heats they produce 1st 12.6, 2nd 13.1 3rd 12.7 and 4th 13.2. All times are rounded up to the next tenth and you should consider aspects of precision and rounding at every stage of this question.

i) What is wrong?                                                                                                         [1]

You must assume that the longer times imply the timer being closer to the starter’s intended moment, fitting (matching) the starting pistol more precisely. You can assume that the four contestants crossed the line at more or less equal spacing.

ii) What do you suggest the recorded times should be? Explain your decisions with sufficient mathematics and give the same precision as their results have.                                    [5]

iii) The other heat of the same age group (and similar times) included two runners about a metre apart, but the clock shows a gap of 0.5 seconds.
What would you expect the gap to be?                                                                           [3]

iv) In the 800 m, with eight runners in each race the times are typically between 2:40 and 3:00. What would you expect half a second gap and
(v) a metre gap to represent at this distance?                                                               [2,2]

vi) Make four suggestions how they might improve their process, but not the equipment, of taking timings.                                                                                                                 [4]



2.  In the long jump the take-off and landing are on the same level.

i)  At what angle would you recommend attempting to jump from the board?
Explain.                                                                                                                                [3]

ii) If you manage 13 secs for 100m, what is your expected speed at the take-off board?                                                                                                                [2]

iii) If you can do a standing jump of 60cm vertically, what speed is your potential vertical component?                                                                                                                         [2]

iv)  Combining these, what is your take-off velocity?                                                          [2]

v)  If you assume that you can convert all of your basic speed (from (ii)) to the angle you gave in (i), what distance would you expect to jump? Consider this as your maximum possible.  [5]

vi)  If you ran at speed |v| and jumped at angle ø, what distance would you expect?        [2]



3. Girls throw a 4kg shot. The height at which girls can reach to release the shot when intending to throw at 45° is assumed to be 190cm ± 10cm.

i) Draw a particle diagram for the shot in flight. State your general mechanical assumptions.[4]

ii) If a good throw is 9m from 190cm, what is the speed of release?                                        [5]

iii) State the expected throw if conditions are the same but the thrower is 10cm taller            [3]



4.  One of your fellow students is considering running one of the sprints. His first modelling assumes a target time of 29 seconds for the 200m.

i) His first model is v=k. Find k                                                                                               [3]

ii) His second model allows for some acceleration, which he thinks is complete at 25metres. By assuming, as he does, that there is no slowing once maximum speed is reached, find how much faster his basic speed must be [compared to that in (i)] to achieve the target time.   [3]

iii) His third model is mildly better, recognising that there is a peak during the race, and follows the form v= k*(32-t)*t where you can find k by integration to be around 0.04.
Find k to 4 sig.fig.                                                                                                                   [3]

iv)  Show when this model breaks down and find the maximum speed. Comment.             [5]

v)  An altogether better model is of the form …………(you write it)……….                           [3]


                                                                         [60]

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