7 A tractor travels in a straight line from a point \(A\) to a point \(B\). The velocity of the tractor is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\).
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The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for
(a) the distance \(A B\),
(b) the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).- The actual velocity of the tractor is given by \(v = 0.04 t - 0.00005 t ^ { 2 }\) for \(0 \leqslant t \leqslant 800\).
(a) Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i).
For the interval \(0 \leqslant t \leqslant 400\), the approximate velocity of the tractor in part (i) is denoted by \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(b) Express \(v _ { 1 }\) in terms of \(t\) and hence show that \(v _ { 1 } - v = 0.00005 ( t - 200 ) ^ { 2 } - 1\).
(c) Deduce that \(- 1 \leqslant v _ { 1 } - v \leqslant 1\).