Edexcel Paper 3 (Paper 3) 2020 October

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A brick \(P\) of mass \(m\) is placed on the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\)
Brick \(P\) is in equilibrium and on the point of sliding down the plane.
Brick \(P\) is modelled as a particle.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on brick \(P\)
2. show that \(\mu = \frac { 3 } { 4 }\) For parts (c) and (d), you are not required to do any further calculations.
   Brick \(P\) is now removed from the plane and a much heavier brick \(Q\) is placed on the plane. The coefficient of friction between \(Q\) and the plane is also \(\frac { 3 } { 4 }\)
3. Explain briefly why brick \(Q\) will remain at rest on the plane. Brick \(Q\) is now projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of the plane.
   Brick \(Q\) is modelled as a particle.
   Using the model,
4. describe the motion of brick \(Q\), giving a reason for your answer.

1. A particle \(P\) moves with acceleration \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)

At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0 , P\) passes through the origin \(O\).
   At time \(t = T\) seconds, where \(T > 0\), the particle \(P\) passes through the point \(A\).
   The position vector of \(A\) is ( \(\lambda \mathbf { i } - 4.5 \mathbf { j }\) )m relative to \(O\), where \(\lambda\) is a constant.
2. Find the value of \(T\).
3. Hence find the value of \(\lambda\)

1. 1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration a \(\mathrm { ms } ^ { - 2 }\) is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) when \(t = 4\)
2. Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to i
   (ii) At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

4. \begin{figure}[h] \end{figure} A ladder \(A B\) has mass \(M\) and length \(6 a\).
The end \(A\) of the ladder is on rough horizontal ground.
The ladder rests against a fixed smooth horizontal rail at the point \(C\).
The point \(C\) is at a vertical height \(4 a\) above the ground.
The vertical plane containing \(A B\) is perpendicular to the rail.
The ladder is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 4 } { 5 }\), as shown in Figure 1.
The coefficient of friction between the ladder and the ground is \(\mu\).
The ladder rests in limiting equilibrium.
The ladder is modelled as a uniform rod.
Using the model,

1. show that the magnitude of the force exerted on the ladder by the rail at \(C\) is \(\frac { 9 M g } { 25 }\)
2. Hence, or otherwise, find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A small ball is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The point \(O\) is 25 m vertically above the point \(N\) which is on horizontal ground. The ball is projected at an angle of \(45 ^ { \circ }\) above the horizontal.
The ball hits the ground at a point \(A\), where \(A N = 100 \mathrm {~m}\), as shown in Figure 2 .
The motion of the ball is modelled as that of a particle moving freely under gravity.
Using this initial model,

1. show that \(U = 28\)
2. find the greatest height of the ball above the horizontal ground \(N A\). In a refinement to the model of the motion of the ball from \(O\) to \(A\), the effect of air resistance is included. This refined model is used to find a new value of \(U\).
3. How would this new value of \(U\) compare with 28, the value given in part (a)?
4. State one further refinement to the model that would make the model more realistic. \section*{" " \(_ { \text {" } } ^ { \text {" } }\) " "}