Edexcel AS Paper 2 (AS Paper 2) 2021 November

1. At time \(t = 0\), a small stone is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point \(A\).

At time \(t = T\) seconds, the stone passes through \(A\), moving downwards.
The stone is modelled as a particle moving freely under gravity throughout its motion.
Using the model,

1. find the value of \(T\),
2. find the total distance travelled by the stone in the first 4 seconds of its motion.
3. State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

1. A particle \(P\) moves along a straight line.

At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.

1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)

3. \begin{figure}[h] \end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit the pulley. The balls are modelled as particles.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Air resistance is modelled as being negligible.
Using this model,

1. 1. write down an equation of motion for \(P\),
   2. write down an equation of motion for \(Q\),
2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
4. State how this would affect the accelerations of the particles.

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