Edexcel AS Paper 2 (AS Paper 2) 2023 June

1. \begin{figure}[h] \end{figure} Two children, Pat \(( P )\) and Sam \(( S )\), run a race along a straight horizontal track.
Both children start from rest at the same time and cross the finish line at the same time.
In a model of the motion:
Pat accelerates at a constant rate from rest for 5 s until reaching a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and then maintains a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\) until crossing the finish line. Sam accelerates at a constant rate of \(1 \mathrm {~ms} ^ { - 2 }\) from rest until reaching a speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains a constant speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until crossing the finish line. Both children take 27.5 s to complete the race.
The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together. Using the model,

1. explain why the areas under the two graphs are equal,
2. find the acceleration of Pat during the first 5 seconds,
3. find, in metres, the length of the race,
4. find the value of \(X\), giving your answer to 3 significant figures.

1. A small stone is projected vertically upwards with speed \(39.2 \mathrm {~ms} ^ { - 1 }\) from a point \(O\).

The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10s later. Using the model, find

1. the height of \(O\) above the ground,
2. the total length of time for which the speed of the stone is less than or equal to \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. State one refinement that could be made to the model that would make your answer to part (a) more accurate.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A fixed point \(O\) lies on a straight line.
A particle \(P\) moves along the straight line such that at time \(t\) seconds, \(t \geqslant 0\), after passing through \(O\), the velocity of \(P , v \mathrm {~ms} ^ { - 1 }\), is modelled as $$v = 15 - t ^ { 2 } - 2 t$$

1. Verify that \(P\) comes to instantaneous rest when \(t = 3\)
2. Find the magnitude of the acceleration of \(P\) when \(t = 3\)
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

4. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.
The rope is horizontal and parallel to the direction of motion of the car.

• The resistance to motion of the car is modelled as a constant force of magnitude \(2 R\) newtons
• The resistance to motion of the trailer is modelled as a constant force of magnitude \(R\) newtons
• The rope is modelled as being light and inextensible
• The acceleration of the car is modelled as \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N . Using the model,

1. find the value of \(a\) In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be \(a _ { 1 } \mathrm {~ms} ^ { - 2 }\)
2. State how the value of \(a _ { 1 }\) compares with the value of \(a\)
3. State one limitation of the model used for the resistance to motion of the car.