10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\).
Rory proposes a model in which the acceleration is constant until time \(T\) seconds.
- Sketch the velocity-time graph for Rory's run using this model.
- Calculate \(T\).
- Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
- Use this model to find the time taken for Rory to run the first 4 m .
Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
- Sketch a velocity-time graph that represents Rory's run using this refined model.
- State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)