Edexcel M3 (Mechanics 3)

1. A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, \(s\) metres, of a point \(P\) on the airbag \(t\) seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

1. find, correct to the nearest centimetre, the maximum displacement of \(P\),
2. find the initial velocity of \(P\),
3. find the acceleration of \(P\) in terms of \(t\).
4. Explain why this model is unlikely to be realistic.

2. A particle \(P\) is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point \(A\). \begin{figure}[h] \end{figure} When a horizontal force of magnitude 20 N is applied to \(P\), it rests in equilibrium with the string making an angle of \(30 ^ { \circ }\) with the vertical and \(A P = 1.2 \mathrm {~m}\) as shown in Figure 1.

1. Find the tension in the string.
2. Find the elastic potential energy stored in the string.

3. \begin{figure}[h] \end{figure} A particle of mass \(m\) is suspended at a point \(A\) vertically below a fixed point \(O\) by a light inextensible string of length \(a\) as shown in Figure 2. The particle is given a horizontal velocity \(u\) and subsequently moves along a circular arc until it reaches the point \(B\) where the string becomes slack. Given that the point \(B\) is at a height \(\frac { 1 } { 2 } a\) above the level of \(O\),

1. show that \(\angle B O A = 120 ^ { \circ }\),
2. show that \(u ^ { 2 } = \frac { 7 } { 2 } g a\).

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs \(6 \frac { 1 } { 4 }\) hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

1. write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
2. Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
   (6 marks)
   Given that a ship is not ready to enter the harbour until 5 p.m.,
3. find, to the nearest minute, how long the ship must wait before it can enter the harbour.

5. (a) Use integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is \(\frac { 3 } { 4 } h\) from the vertex of the cone.
(6 marks) \begin{figure}[h] \end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius \(r\) so that the remaining solid consists of a hemisphere of radius \(r\) and a cone of height \(r\) and base radius \(r\) as shown in Figure 3.
(b) Find the distance of the centre of mass of the paperweight from its vertex.
(7 marks)

6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is \(\frac { 2 } { 5 }\).

1. Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
   (5 marks)
   The owner of the track decides to bank the corner at an angle of \(25 ^ { \circ }\) in order to enable the cars to travel more quickly.
2. Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
   (8 marks)

7. A particle is travelling along the \(x\)-axis. At time \(t = 0\), the particle is at \(O\) and it travels such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a distance \(x\) metres from \(O\) is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is \(a \mathrm {~ms} ^ { - 2 }\).

1. Show that \(a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }\).
   (4 marks) The points \(A\) and \(B\) lie on the \(x\)-axis. Given that the particle travels \(d\) metres from \(O\) to \(A\) in \(T\) seconds and 4 metres from \(A\) to \(B\) in 9 seconds,
2. show that \(d = 1.5\),
3. find \(T\).