Two trains, \(P\) and \(Q\), move on horizontal parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , P\) starts off and moves with constant acceleration for 10 s up to a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 20\), where \(t\) is measured in seconds, train \(Q\) starts to move in the same direction as \(P\). Train \(Q\) accelerates with the same initial constant acceleration as \(P\), up to a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(Q\) overtakes \(P\) at time \(t = T\), after both trains have reached their constant speeds.
Sketch, on the same axes, the speed-time graphs of both trains for \(0 \leqslant t \leqslant T\).
Find the value of \(t\) at the instant when both trains are moving at the same speed.