OCR M1 (Mechanics 1) Specimen

1
\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_99_812_310_635} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude \(E\) newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N , and the acceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(E\).

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30 ^ { \circ }\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at an angle \(\theta ^ { \circ }\) to the force of magnitude 8 N , as shown in Fig. 2. Find \(R\) and \(\theta\).

3 A particle is projected vertically upwards, from the ground, with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring air resistance, find

1. the maximum height reached by the particle,
2. the speed of the particle when it is 30 m above the ground,
3. the time taken for the particle to fall from its highest point to a height of 30 m ,
4. the length of time for which the particle is more than 30 m above the ground. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_569_1132_258_516} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s . The woman runs at a constant speed throughout. Fig. 1 shows the \(( t , v )\) graph for the woman.
5. Find the total distance run by the woman.
6. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_424_1135_1233_513} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} At time \(t = 0\), a child also starts to move, from \(A\), along \(A B\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s . Fig. 2 shows the ( \(t , v\) ) graph for the child; it consists of two straight line segments.
7. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s .
8. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant.

5 A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(- \frac { 1 } { 10 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\).
2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\).
3. Find the displacement of \(P\) from \(O\) when \(t = 10\).
4. Find the speed with which the particle returns to \(O\).

6
\includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres \(A , B\) and \(C\) have masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) (see diagram). Air resistance may be ignored.

1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed with which \(B\) starts to move.
2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\).
4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers.

7 A sledge of mass 25 kg is on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2 .

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_289_488_1493_849} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after being pulled for 10 s . Ignoring air resistance, find the tension in the cable.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_291_490_2149_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane.