Edexcel Paper 3 (Paper 3) 2022 June

1. \hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) where

$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$

1. Find the speed of \(P\) at time \(t = 2\) seconds.
2. Find an expression, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\), for the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\) At time \(t = 4\) seconds, the position vector of \(P\) is ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
3. Find the position vector of \(P\) at time \(t = 1\) second.

2. \begin{figure}[h] \end{figure} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
A small block \(B\) of mass 5 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 1. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane. The block \(B\) is modelled as a particle.
The magnitude of the normal reaction of the plane on \(B\) is 68.6 N .
Using the model,

1. 1. find the magnitude of the frictional force acting on \(B\),
   2. state the direction of the frictional force acting on \(B\). The horizontal force of magnitude \(X\) newtons is now removed and \(B\) moves down the plane. Given that the coefficient of friction between \(B\) and the plane is 0.5
2. find the acceleration of \(B\) down the plane.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]

A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )

1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
   Given that \(\lambda = 2\)
2. find the length of \(A B\).

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(M\) and length \(2 a\)
A particle of mass \(2 M\) is attached to the rod at the point \(C\), where \(A C = 1.5 a\)
The rod rests with its end \(A\) on rough horizontal ground.
The rod is held in equilibrium at an angle \(\theta\) to the ground by a light string that is attached to the end \(B\) of the rod. The string is perpendicular to the rod, as shown in Figure 2.

1. Explain why the frictional force acting on the rod at \(A\) acts horizontally to the right on the diagram. The tension in the string is \(T\)
2. Show that \(T = 2 M g \cos \theta\) Given that \(\cos \theta = \frac { 3 } { 5 }\)
3. show that the magnitude of the vertical force exerted by the ground on the rod at \(A\) is \(\frac { 57 M g } { 25 }\) The coefficient of friction between the rod and the ground is \(\mu\)
   Given that the rod is in limiting equilibrium,
4. show that \(\mu = \frac { 8 } { 19 }\)

5. \begin{figure}[h] \end{figure} A golf ball is at rest at the point \(A\) on horizontal ground.
The ball is hit and initially moves at an angle \(\alpha\) to the ground.
The ball first hits the ground at the point \(B\), where \(A B = 120 \mathrm {~m}\), as shown in Figure 3.
The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using this model,

1. show that \(U ^ { 2 } \sin \alpha \cos \alpha = 588\) The ball reaches a maximum height of 10 m above the ground.
2. Show that \(U ^ { 2 } = 1960\) In a refinement to the model, the effect of air resistance is included.
   The motion of the ball, from \(A\) to \(B\), is now modelled as that of a particle whose initial speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This refined model is used to calculate a value for \(V\)
3. State which is greater, \(U\) or \(V\), giving a reason for your answer.
4. State one further refinement to the model that would make the model more realistic.