OCR FM1 AS (Further Mechanics 1 AS) 2021 June

A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW.

1. The car can travel at constant speed \(v \text{ ms}^{-1}\) along the road. Find \(v\). [2]
2. Find the acceleration of the car at an instant when its speed is \(30 \text{ ms}^{-1}\). [3]

A particle \(P\) of mass 5.6 kg is attached to one end of a light rod of length 2.1 m. The other end of the rod is freely hinged to a fixed point \(O\). The particle is initially at rest directly below \(O\). It is then projected horizontally with speed \(5 \text{ ms}^{-1}\). In the subsequent motion, the angle between the rod and the downward vertical at \(O\) is denoted by \(\theta\) radians, as shown in the diagram. \includegraphics{figure_2}

1. Find the speed of \(P\) when \(\theta = \frac{1}{4}\pi\). [4]
2. Find the value of \(\theta\) when \(P\) first comes to instantaneous rest. [2]

A particle of mass \(m\) moves in a straight line with constant acceleration \(a\). Its initial and final velocities are \(u\) and \(v\) respectively and its final displacement from its starting position is \(s\). In order to model the motion of the particle it is suggested that the velocity is given by the equation $$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$ where \(p\) and \(q\) are dimensionless constants.

1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent. [2]
2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\). [4]
3. By considering the case where \(s = 0\), determine the value of \(p\). [1]
4. By multiplying both sides of the equation by \(\frac{1}{2}m\), and using the numerical values of \(\alpha\), \(\beta\) and \(\gamma\), determine the value of \(q\). [2]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]