OCR M1 (Mechanics 1) 2011 January

1 Two particles \(P\) and \(Q\) are projected directly towards each other on a smooth horizontal surface. \(P\) has mass 0.5 kg and initial speed \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(Q\) has mass 0.8 kg and initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a collision between \(P\) and \(Q\), the speed of \(P\) is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the direction of its motion is reversed. Calculate

1. the change in the momentum of \(P\),
2. the speed of \(Q\) after the collision.

2
\includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-02_597_885_676_630} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate

1. \(\alpha\),
2. \(F\).

3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.

1. Calculate the speed of the particle when it strikes the ground.
2. Calculate the time after projection when the particle reaches the ground.
3. Sketch on separate diagrams
   (a) the \(( t , v )\) graph,
   (b) the \(( t , x )\) graph,
   representing the motion of the particle.

4
\includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_156_1141_258_502} A block \(B\) of mass 0.8 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string inclined at \(10 ^ { \circ }\) to the horizontal. They are pulled across a horizontal surface with acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), by a horizontal force of 2 N applied to \(B\) (see diagram).

1. Given that contact between \(B\) and the surface is smooth, calculate the tension in the string.
2. Calculate the coefficient of friction between \(P\) and the surface.

5
\includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614}
\(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.

1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
   (b) Calculate \(\theta\).
   (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
2. Calculate the speeds of the particles immediately before they collide.

6 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = t ^ { 2 } - 9\). The particle travels in a straight line and passes through a fixed point \(O\) when \(t = 2\).

1. Find the displacement of the particle from \(O\) when \(t = 0\).
2. Calculate the distance the particle travels from its position at \(t = 0\) until it changes its direction of motion.
3. Calculate the distance of the particle from \(O\) when the acceleration of the particle is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A particle \(P\) of mass 0.6 kg is projected up a line of greatest slope of a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves with deceleration \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and comes to rest before reaching the top of the plane.

1. Calculate the frictional force acting on \(P\), and the coefficient of friction between \(P\) and the plane.
2. Find the magnitude of the contact force exerted on \(P\) by the plane and the angle between the contact force and the upward direction of the line of greatest slope,
   (a) when \(P\) is in motion,
   (b) when \(P\) is at rest.

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7(ii) (b

(ii) (b)

\section*{OCR} RECOGNISING ACHIEVEMENT