OCR M1 (Mechanics 1) 2015 June

1 A particle \(P\) is projected vertically downwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 30 m above the ground.

1. Calculate the speed of \(P\) when it reaches the ground.
2. Find the distance travelled by \(P\) in the first 0.4 s of its motion.
3. Calculate the time taken for \(P\) to travel the final 15 m of its descent.

2
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_138_1118_680_463} Three particles \(P , Q\) and \(R\) with masses \(0.4 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are moving along the same straight line on a smooth horizontal surface. \(P\) and \(Q\) are moving towards each other with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(R\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the same direction as \(Q\) (see diagram).

1. Immediately after the collision between \(P\) and \(Q\) their directions of motion have been reversed, but their speeds are unchanged. Calculate \(u\). The next collision is between \(Q\) and \(R\). After the collision between \(Q\) and \(R\), particle \(Q\) is at rest and \(R\) has speed \(9 \mathrm {~ms} ^ { - 1 }\).
2. Calculate \(m\).
   \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_547_1506_1521_251} Two travellers \(A\) and \(B\) make the same journey on a long straight road. Each traveller walks for part of the journey and rides a bicycle for part of the journey. They start their journeys at the same instant, and they end their journeys simultaneously after travelling for \(T\) hours. \(A\) starts the journey cycling at a steady \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) for 1 hour. \(A\) then leaves the bicycle at the side of the road, and completes the journey walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \(B\) begins the journey walking at a steady \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). When \(B\) finds the bicycle where \(A\) left it, \(B\) cycles at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) to complete the journey (see diagram).

1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
2. Find \(T\).
3. Calculate the distance \(A\) and \(B\) each travel.

4
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.

5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:

1. \(\theta = 0\);
2. \(\theta = 20\);
3. \(\theta = 70\);
4. \(\theta = 90\).

6 A particle \(P\) moves in a straight line on a horizontal surface. \(P\) passes through a fixed point \(O\) on the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).

1. Calculate the velocity of \(P\) when \(t = 3\).
2. Find the distance \(O P\) when \(t = 3\). A second particle \(Q\), having the same mass as \(P\), moves along the same straight line. The displacement of \(Q\) from \(O\) is \(\left( k - 2 t ^ { 3 } \right) \mathrm { m }\), where \(k\) is a constant. When \(t = 3\) the particles collide and coalesce.
3. Find the value of \(k\).
4. Find the common velocity of the particles immediately after their collision.

7
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424}
\(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).

1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
   (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}