OCR M1 (Mechanics 1) 2006 June

1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass \(m \mathrm {~kg}\) and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(m\).

2
\includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-2_620_711_543_717} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at right angles to the force of magnitude 2.5 N (see diagram).

1. Show that \(\theta = 22.6 ^ { \circ }\), correct to 3 significant figures.
2. Find the value of \(R\).

3 A man travels 360 m along a straight road. He walks for the first 120 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), runs the next 180 m at \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then walks the final 60 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \(( t , x )\) graph for the journey, showing the values of \(t\) for which \(x = 120,300\) and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
2. Draw a dotted line on your ( \(t , x\) ) graph to represent the woman's journey.
3. Calculate the value of \(t\) at which the man overtakes the woman.

4 A cyclist travels along a straight road. Her velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds after starting from a point \(O\), is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10
& v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$

1. Find the displacement of the cyclist from \(O\) when \(t = 10\).
2. Show that, for \(t \geqslant 10\), the displacement of the cyclist from \(O\) is given by the expression \(0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5\).
3. Find the time when the acceleration of the cyclist is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Hence find the displacement of the cyclist from \(O\) when her acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5 A block of mass \(m \mathrm {~kg}\) is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .

1. When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of \(m\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-3_312_711_1244_758} When a force of magnitude \(P \mathrm {~N}\) acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
   (a) the value of \(\alpha\) in degrees,
   (b) the value of \(P\).

6
\includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-4_314_997_267_577} A train of total mass 80000 kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E , A\) and \(B\) are \(10500 \mathrm {~N} , 3000 \mathrm {~N}\) and 1500 N respectively (see diagram).

1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
2. Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling \(Y\) is zero.
4. Find the mass of \(B\).
5. Show that the coupling \(X\) exerts a forward force of magnitude 1500 N on the engine.

7 A particle of mass 0.1 kg is at rest at a point \(A\) on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The particle is given an initial velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .

1. Find the coefficient of friction between the particle and the plane.
2. Show that, after coming to instantaneous rest, the particle moves down the plane.
3. Find the speed with which the particle passes through \(A\) during its downward motion.