Two vehicles: overtaking or meeting (graph-based)

3 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_556_974_1548_587} The diagram shows the velocity-time graphs for the motion of two cyclists \(P\) and \(Q\), who travel in the same direction along a straight path. Both cyclists start from rest at the same point \(O\) and both accelerate at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Both then continue at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(Q\) starts his journey \(T\) seconds after \(P\).

1. Show in a sketch of the diagram the region whose area represents the displacement of \(P\), from \(O\), at the instant when \(Q\) starts. Given that \(P\) has travelled 16 m at the instant when \(Q\) starts, find
2. the value of \(T\),
3. the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(10 \mathrm {~ms} ^ { - 1 }\).

3 \(\mathrm { v } \left( \mathrm { ms } ^ { - 1 } \right)\) \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_449_1121_1500_440}
not to scale The diagram shows the \(( t , v )\) graphs for two athletes, \(A\) and \(B\), who run in the same direction in the same straight line while they exchange the baton in a relay race. \(A\) runs with constant velocity \(10 \mathrm {~ms} ^ { - 1 }\) until he decelerates at \(5 \mathrm {~ms} ^ { - 2 }\) and subsequently comes to rest. \(B\) has constant acceleration from rest until reaching his constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The baton is exchanged 2 s after \(B\) starts running, when both athletes have speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) is 1 m ahead of \(A\).

1. Find the value of \(t\) at which \(A\) starts to decelerate.
2. Calculate the distance between \(A\) and \(B\) at the instant when \(B\) starts to run.

8 Two trains, \(A\) and \(B\), are moving on straight horizontal tracks which run alongside each other and are parallel. The trains both move with constant acceleration. At time \(t = 0\), the fronts of the trains pass a signal. The velocities of the trains are shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-18_633_1077_475_424}

1. Find the distance between the fronts of the two trains when they have the same velocity and state which train has travelled further from the signal.
2. Find the time when \(A\) has travelled 9 metres further than \(B\).
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-20_2288_1707_221_153}

\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.

1. Calculate the deceleration of car \(A\). [2]
2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]