Edexcel Paper 3 (Paper 3) 2023 June

1. A car is initially at rest on a straight horizontal road.

The car then accelerates along the road with a constant acceleration of \(3.2 \mathrm {~ms} ^ { - 2 }\)
Find

1. the speed of the car after 5 s ,
2. the distance travelled by the car in the first 5 s .

2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.

1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where

$$\mathbf { v } = \left( t ^ { 2 } - 3 t + 7 \right) \mathbf { i } + \left( 2 t ^ { 2 } - 3 \right) \mathbf { j }$$ Find

1. the speed of \(P\) at time \(t = 0\)
2. the value of \(t\) when \(P\) is moving parallel to \(( \mathbf { i } + \mathbf { j } )\)
3. the acceleration of \(P\) at time \(t\) seconds
4. the value of \(t\) when the direction of the acceleration of \(P\) is perpendicular to \(\mathbf { i }\)

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

A particle \(P\) is moving on a smooth horizontal plane.
The particle has constant acceleration \(( 2.4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0 , P\) passes through the point \(A\).
At time \(t = 5 \mathrm {~s} , P\) passes through the point \(B\).
The velocity of \(P\) as it passes through \(A\) is \(( - 16 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the speed of \(P\) as it passes through \(B\). The position vector of \(A\) is \(( 44 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\).
   At time \(t = T\) seconds, where \(T > 5 , P\) passes through the point \(C\).
   The position vector of \(C\) is \(( 4 \mathbf { i } + c \mathbf { j } ) \mathrm { m }\).
2. Find the value of \(T\).
3. Find the value of \(c\).

5. \begin{figure}[h] \end{figure} A small ball is projected with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on horizontal ground. After moving for \(T\) seconds, the ball passes through the point \(A\). The point \(A\) is 40 m horizontally and 20 m vertically from the point \(O\), as shown in Figure 2. The motion of the ball from \(O\) to \(A\) is modelled as that of a particle moving freely under gravity. Given that the ball is projected at an angle \(\alpha\) to the ground, use the model to

1. show that \(T = \frac { 10 } { 7 \cos \alpha }\)
2. show that \(\tan ^ { 2 } \alpha - 4 \tan \alpha + 3 = 0\)
3. find the greatest possible height, in metres, of the ball above the ground as the ball moves from \(O\) to \(A\). The model does not include air resistance.
4. State one other limitation of the model.

6. \begin{figure}[h] \end{figure} A \(\operatorname { rod } A B\) has mass \(M\) and length \(2 a\).
The rod has its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the ground, as shown in Figure 3.
The rod is at rest in limiting equilibrium.

1. State the direction (left or right on Figure 3 above) of the frictional force acting on the \(\operatorname { rod }\) at \(A\). Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at \(B\) is \(S\).
   In an initial model, the rod is modelled as being uniform.
   Use this initial model to answer parts (b), (c) and (d).
2. By taking moments about \(A\), show that $$S = \frac { 1 } { 2 } M g \cot \theta$$ The coefficient of friction between the rod and the ground is \(\mu\)
   Given that \(\tan \theta = \frac { 3 } { 4 }\)
3. find the value of \(\mu\)
4. find, in terms of \(M\) and \(g\), the magnitude of the resultant force acting on the rod at \(A\). In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to \(B\) than it is to \(A\). A new value for \(S\) is calculated using this new model, with \(\tan \theta = \frac { 3 } { 4 }\)
5. State whether this new value for \(S\) is larger, smaller or equal to the value that \(S\) would take using the initial model. Give a reason for your answer.