Edexcel M1 (Mechanics 1) 2016 October

1. Two particles, \(P\) and \(Q\), have masses \(2 m\) and \(3 m\) respectively. They are moving towards each other, in opposite directions, along the same straight line, on a smooth horizontal plane. The particles collide. Immediately before they collide the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). In the collision the magnitude of the impulse exerted on \(P\) by \(Q\) is \(5 m u\).
   1. Find the speed of \(P\) immediately after the collision.
   2. State whether the direction of motion of \(P\) has been reversed by the collision.
   3. Find the speed of \(Q\) immediately after the collision.
      

2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] Three forces, \(( - 10 \mathbf { i } + a \mathbf { j } ) \mathrm { N } , ( b \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( 2 a \mathbf { i } + 7 \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a particle \(P\) of mass 3 kg . The acceleration of \(P\) is \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

1. Find the value of \(a\) and the value of \(b\). At time \(t = 0\) seconds the speed of \(P\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at time \(t = 4\) seconds the velocity of \(P\) is \(( 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the value of \(u\).

3. \begin{figure}[h] \end{figure} A plank \(A B\) has length 8 m and mass 12 kg . The plank rests on two supports. One support is at \(C\), where \(A C = 3 \mathrm {~m}\) and the other support is at \(D\), where \(A D = x\) metres. A block of mass 3 kg is placed on the plank at \(B\), as shown in Figure 1. The plank rests in equilibrium in a horizontal position. The magnitude of the force exerted on the plank by the support at \(D\) is twice the magnitude of the force exerted on the plank by the support at \(C\). The plank is modelled as a uniform rod and the block is modelled as a particle. Find the value of \(x\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\) ]

A particle \(P\) is moving with velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\) hours, the position vector of \(P\) is \(( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).

1. Find an expression for \(\mathbf { p }\) in terms of \(t\). The point \(A\) has position vector ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) km.
2. Find the position vector of \(P\) when \(P\) is due west of \(A\). Another particle \(Q\) is moving with velocity \([ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }\) where \(b\) is a constant. Given that the particles are moving along parallel lines,
3. find the value of \(b\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. Car \(A\) is moving with uniform acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\) and car \(B\) is moving with uniform acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant when \(B\) is 200 m behind \(A\), the speed of \(A\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(44 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(B\) when it overtakes \(A\).
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7. A train moves on a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(1 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train maintains this speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next \(T\) seconds before slowing down with constant deceleration \(0.5 \mathrm {~ms} ^ { - 2 }\), coming to rest at \(B\). The journey from \(A\) to \(B\) takes 180 s and the distance between the stations is 4800 m .

1. Sketch a speed-time graph for the motion of the train from \(A\) to \(B\).
2. Show that \(T = 180 - 3 V\).
3. Find the value of \(V\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion before \(Q\) strikes the ground, find the tension in the string.
2. Find the value of \(\mu\).

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