CAIE Further Paper 3 (Further Paper 3) 2024 November

A particle \(P\) is projected with speed \(u\text{ms}^{-1}\) at an angle \(\tan^{-1}2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56\text{m}\) horizontally from \(O\), it is at a vertical height \(H\text{m}\) above the plane. When \(P\) has travelled a distance \(84\text{m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H\text{m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan\theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]

A particle \(P\) of mass \(m\text{kg}\) is attached to one end of a light elastic string of natural length \(2\text{m}\) and modulus of elasticity \(2mg\text{N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\text{m}\) below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]

1. Hence find the speed of \(P\) when it is \(2\text{m}\) below \(O\). [2]

\includegraphics{figure_4} An object is formed by removing a cylinder of radius \(\frac{2}{3}a\) and height \(kh\) (\(k < 1\)) from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points \(A\) and \(B\) are the opposite ends of a diameter of the upper face of the object (see diagram).

1. Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(AB\). [4]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

1. Find the magnitude of \(F\) when \(t = 3\). [3]

\includegraphics{figure_6} A particle \(P\) of mass \(0.05\text{kg}\) is attached to one end of a light inextensible string of length \(1\text{m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04\text{kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8\text{m}\) with angular speed \(\omega\text{rads}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4\text{m}\) also with angular speed \(\omega\text{rads}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).

1. Find the tension in the string \(OP\). [3]

1. Find the value of \(\omega\). [3]

1. Find the value of \(\beta\). [2]

A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).

1. Show that \(e\tan^2\alpha = 1\). [4]

In its subsequent motion, the greatest height reached by \(P\) above \(A\) is \(\frac{3}{10}\) of the vertical height of \(A\) above the horizontal plane.

1. Find the value of \(e\). [6]