Two small children, Ajaz and Beth, are running a $$\(100 \mathrm{~m}\)$$ race along a straight horizontal track. They both start from rest, leaving the start line at the same time. Ajaz accelerates at $$\(0.8 \mathrm{~ms}^{-2}\)$$ up to a speed of $$\(4 \mathrm{~m} \mathrm{~s}^{-1}\)$$ and then maintains this speed until he crosses the finish line. Beth accelerates at $$\(1 \mathrm{~ms}^{-2}\)$$ for $$\(T\)$$ seconds and then maintains a constant speed until she crosses the finish line. Ajaz and Beth cross the finish line at the same time. (a) Sketch, on the same axes, a speed-time graph for each child, from the instant when they leave the start line to the instant when they cross the finish line. (3) (b) Find the time taken by Ajaz to complete the race. (4) (c) Find the value of $$\(T\)$$ (4) (d) Find the difference in the speeds of the two children as they cross the finish line. (2)

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