CAIE Further Paper 3 (Further Paper 3) 2021 November

1 A particle is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane. The particle moves freely under gravity.

1. Write down the horizontal and vertical components of the velocity of the particle at time \(T\) after projection.
   At time \(T\) after projection, the direction of motion of the particle is perpendicular to the direction of projection.
2. Express \(T\) in terms of \(u , g\) and \(\alpha\).
3. Deduce that \(\mathrm { T } > \frac { \mathrm { u } } { \mathrm { g } }\).

2 A light spring \(A B\) has natural length \(a\) and modulus of elasticity 5 mg . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(k m\) is moving with speed \(\sqrt { 4 \mathrm { ga } }\) along the horizontal surface towards \(P\) in the direction \(B A\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac { 1 } { 5 } a\). Find the value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-04_307_1088_274_470} Particles \(A\) and \(B\), of masses \(m\) and \(3 m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram).

1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
2. Find an expression for \(v\) in terms of \(a\) and \(g\).
   \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-06_597_803_258_625} An object is formed by removing a solid cylinder, of height \(k a\) and radius \(\frac { 1 } { 2 } a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(A B\) is a diameter of the circular face of the hemisphere (see diagram).

1. Show that the distance of the centre of mass of the object from \(A B\) is \(\frac { 3 \mathrm { a } \left( 2 - \mathrm { k } ^ { 2 } \right) } { 2 ( 8 - 3 \mathrm { k } ) }\).
   When the object is freely suspended from the point \(A\), the line \(A B\) makes an angle \(\theta\) with the downward vertical, where \(\tan \theta = \frac { 7 } { 18 }\).
2. Find the possible values of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-08_494_903_267_525} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 3 } { 2 } m\) respectively. The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision \(A\) 's direction of motion is along the line of centres, and \(B\) 's direction of motion makes an angle of \(60 ^ { \circ }\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\).
3. Find the angle through which the direction of motion of \(B\) is deflected by the collision.
4. Find the loss in the total kinetic energy of the system as a result of the collision.

6 A particle \(P\) of mass 2 kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\).
A force of magnitude \(\left( 8 x - \frac { 128 } { x ^ { 3 } } \right) \mathrm { N }\) acts on \(P\) in the direction \(O P\). When \(\mathrm { t } = 0 , \mathrm { x } = 8\) and \(\mathrm { v } = - 15\).

1. Show that \(\mathrm { v } = - \frac { 2 } { \mathrm { x } } \left( \mathrm { x } ^ { 2 } - 4 \right)\).
2. Find an expression for \(x\) in terms of \(t\).

7 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60 ^ { \circ }\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).

1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\).
2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.