Two-particle meeting or overtaking

10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }

A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$

A particle \(A\) moves in a straight line with constant speed \(10\) m s\(^{-1}\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16\) m s\(^{-1}\) at \(O\) and has a constant deceleration of \(2\) m s\(^{-2}\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) s after \(B\) passes through \(O\). [3]
2. Find the distance between the particles when \(B\) comes to instantaneous rest. [3]
3. Find the minimum distance between the particles. [3]

Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6(t - 3)\) m s\(^{-1}\) and the velocity of \(Q\) after \(t\) seconds is \((10 - 2t)\) m s\(^{-1}\).

1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leq t \leq 5\). [3]
2. Verify that \(P\) and \(Q\) meet after 5 seconds. [4]
3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leq t \leq 5\). [4]

A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m s\(^{-2}\) for a distance of 36 m. The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(AB\) is 210 m.

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]

24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.

\includegraphics{figure_2} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a\) m s\(^{-2}\) for 3.5 s, reaching a speed of 14 m s\(^{-1}\). Car \(A\) then moves with constant speed 14 m s\(^{-1}\).

1. Find the value of \(a\). [2] Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with constant acceleration of 3 m s\(^{-2}\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]
3. Find the value of \(T\). [8]
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\). [1]
5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = T\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. At time \(t = 0\), they are side by side, passing a point \(O\) on the road. Car \(A\) travels at a constant speed of \(30 \text{ m s}^{-1}\). Car \(B\) passes \(O\) with a speed of \(20 \text{ m s}^{-1}\), and has constant acceleration of \(4 \text{ m s}^{-2}\). Find

1. the speed of \(B\) when it has travelled 78 m from \(O\), [2]
2. the distance from \(O\) of \(A\) when \(B\) is 78 m from \(O\), [4]
3. the time when \(B\) overtakes \(A\). [5]

\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]

Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}

1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
   v &= 8 \quad \text{for } t > 4. \end{align} [3]
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]

Two cars, P and Q, are being crashed as part of a film 'stunt'. At the start

• P is travelling directly towards Q with a speed of \(8\) ms\(^{-1}\),
• Q is instantaneously at rest and has an acceleration of \(4\) ms\(^{-2}\) directly towards P.

P continues with the same velocity and Q continues with the same acceleration. The cars collide \(T\) seconds after the start.

1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start. [2]

At the start, P is 90 m from Q.

1. Show that \(T^2 + 4T - 45 = 0\) and hence find \(T\). [5]

\includegraphics{figure_6} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards Q of \(0.5\text{ m s}^{-2}\). Particle Q starts 125 m from P at the same time and has a constant speed of \(10\text{ m s}^{-1}\) away from P. The initial values are shown in Fig. 4.

1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion. [2]
2. How much time does it take for P to catch up with Q and how far does P travel in this time? [5]