CAIE Further Paper 3 (Further Paper 3) 2020 June

A particle \(P\) is projected with speed \(u\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac{2}{3}T\) after projection. [5]

\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]

One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(5mg\), is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac{3}{5}a\).

1. Show that the initial acceleration of \(P\) is \(\frac{3}{5}g\) upwards. [3]
2. Find the speed of \(P\) when the spring first returns to its natural length. [4]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) has sides of length \(10\text{cm}\). The point \(E\) is on \(BC\) with \(EC = 7.5\text{cm}\), and the point \(F\) is on \(DC\) with \(CF = x\text{cm}\). The triangle \(EFC\) is removed from \(ABCD\) (see diagram). The centre of mass of the resulting shape \(ABEFD\) is a distance \(\bar{x}\text{cm}\) from \(CB\) and a distance \(\bar{y}\text{cm}\) from \(CD\).

1. Show that \(\bar{x} = \frac{400 - x^2}{80 - 3x}\) and find a corresponding expression for \(\bar{y}\). [4]

The shape \(ABEFD\) is in equilibrium in a vertical plane with the edge \(DF\) resting on a smooth horizontal surface.

1. Find the greatest possible value of \(x\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants to be determined. [3]

A particle \(P\) is moving along a straight line with acceleration \(3ku - kv\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.

1. Find the time taken for \(P\) to achieve a velocity of \(2u\). [3]
2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2u\). [5]

A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of \(P\) makes an angle \(\alpha\) with the barrier. The coefficient of restitution between \(P\) and the barrier is \(e\). As a result of the impact, the direction of motion of \(P\) is turned through \(90°\).

1. Show that \(\tan^2 \alpha = \frac{1}{e}\). [3]

The particle \(P\) loses two-thirds of its kinetic energy in the impact.

1. Find the value of \(\alpha\) and the value of \(e\). [5]

A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2}ga}\). The particle \(P\) loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\theta = 60°\). [5]
2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\). [5]