科目：进阶数学 Further Mathematics (9231)
年份：2023 冬季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving with equal speeds $u$ on a smooth horizontal surface when they collide. Immediately before the collision, $A$’s direction of motion makes an angle of $60^\circ$ with the line of centres, and $B$’s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $e$.
After the collision, the component of the velocity of $A$ along the line of centres is $v$ and $B$ moves perpendicular to the line of centres. Sphere $A$ now has twice as much kinetic energy as sphere $B$.
(a) Show that $v = \frac{1}{2} u(4\cos \theta - 1)$.
[1]
(b) Find the value of $\cos \theta$.(c) Find the value of $e$. [2]
2 A ball of mass 2 kg is projected vertically downwards with speed $5 \text{ m/s}^{-1}$ through a liquid. At time $ts$ after projection, the velocity of the ball is $v \text{ m/s}^{-1}$ and its displacement from its starting point is $x \text{ m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2v^2 \text{ N}$.
(a) Find an expression for $v$ in terms of $t$. [6](b) Deduce what happens to $v$ for large values of $t$. [1]
A uniform square lamina of side $2a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $AB$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $EP$ is perpendicular to the side $AB$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac{1}{2}$.
Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$. [8]7 © UCLES 2023 9231/31/OON/23 [Turn over]
A light elastic string has natural length $8a$ and modulus of elasticity $5mg$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $AP = BP = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $AB$ it has speed $\sqrt{80ag}$.
(a) Find $L$ in terms of $a$. [5](b) Find the initial acceleration of $P$ in terms of $g$. [3]
5 A particle $P$ is projected with speed $u \text{ ms}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. During its flight $P$ passes through the point which is a horizontal distance $3a$ from $O$ and a vertical distance $\frac{3}{8}a$ above the horizontal plane. It is given that $\tan \theta = \frac{1}{3}$.
(a) Show that $u^2 = 8ag$.
[2]
A particle $Q$ is projected with speed $V \text{ ms}^{-1}$ at an angle $\alpha$ above the horizontal from $O$ at the instant when $P$ is at its highest point. Particles $P$ and $Q$ both land at the same point on the horizontal plane at the same time.
(b) Find $V$ in terms of $a$ and $g$. [7]11 © UCLES 2023 9231/31/OON/23 [Turn over]
6 A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $OQ = x$. The system, of rod and particles, rotates about $O$ in a vertical plane.
At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of $60^\circ$ from the vertical, the speed of $P$ is $2\sqrt{ag}$, and the tensions in the two parts of the rod, $OP$ and $OQ$, have equal magnitudes.
(a) Show that the speed of $Q$ when the rod has turned through an angle of $60^\circ$ from the vertical is $\frac{2x}{3a-x}\sqrt{ag}$.
(b) Find $x$ in terms of $a$.(c) Find $u$ in terms of $a$ and $g$. [4]
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