科目：进阶数学 Further Mathematics (9231)
年份：2024 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
Two smooth uniform spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $\frac{1}{2}u$ respectively. Immediately before the collision, $A$’s direction of motion is along the line of centres, and $B$’s direction of motion makes an angle $\theta$ with the line of centres (see diagram).
As a result of the collision, the direction of motion of $A$ is reversed and its speed is reduced to $\frac{1}{4}u$. The direction of motion of $B$ again makes an angle $\theta$ with the line of centres, but on the opposite side of the line of centres. The speed of $B$ is unchanged.
Find the value of the coefficient of restitution between the spheres. [4]2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $2mg$. A particle $Q$ of mass $km$ is attached to the other end of the string. Particle $P$ lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole $H$ in the table.
Particle $P$ moves in a horizontal circle on the surface of the table with constant speed $\sqrt{\frac{1}{2}ga}$. Particle $Q$ hangs in equilibrium vertically below the hole with $HQ = \frac{1}{4}a$.
(a) Find, in terms of $a$, the extension in the string. [4]
(b) Find the value of $k$. [2]3 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 $O$. When the particle is hanging vertically below $O$, it is projected horizontally with speed $u$ so that it begins to move along a circular path. When $P$ is at the lowest point of its motion, the tension in the string is $T$. When $OP$ makes an angle $\theta$ with the upward vertical, the tension in the string is $S$. (a) Show that $S = T - 3mg(1 + \cos \theta)$. [5]
(b) Given that $u = \sqrt{4ag}$, find the value of $\cos \theta$ when the string goes slack. [2]A light spring of natural length $a$ and modulus of elasticity $kmg$ is attached to a fixed point $O$ on a smooth plane inclined to the horizontal at an angle $\theta$, where $\sin \theta = \frac{3}{4}$. A particle of mass $m$ is attached to the lower end of the spring and is held at the point $A$ on the plane, where $OA = 2a$ and $OA$ is along a line of greatest slope of the plane (see diagram).
The particle is released from rest and is moving with speed $V$ when it passes through the point $B$ on the plane, where $OB = \frac{3}{2}a$. The speed of the particle is $\frac{1}{2}V$ when it passes through the point $C$ on the plane, where $OC = \frac{3}{4}a$.
Find the value of $k$. [7]7 © UCLES 2024 9231/33/M/J/24 [Turn over]
A uniform lamina is in the form of a triangle $OBC$, with $OC = 18a$, $OB = 24a$ and angle $COB = 90^\circ$. The point $A$ on $OB$ is such that $OA = x$ (see diagram). The triangle $OAC$ is removed from the lamina. (a) Find, in terms of $a$ and $x$, the distance of the centre of mass of the remaining object $ABC$ from $OC$. [3]
The object $ABC$ is suspended from $C$. In its equilibrium position, the side $AB$ makes an angle $\theta$ with the vertical, where $\tan \theta = \frac{6}{5}$.
(b) Find $x$ in terms of $a$. [4]6 A particle $P$ is projected with speed $u \text{ ms}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ and moves freely under gravity. After 5 seconds the speed of $P$ is $\frac{3}{4} u$.
(a) Show that $\frac{7}{16} u^2 - 100u \sin \theta + 2500 = 0$. [3](b) It is given that the velocity of $P$ after 5 seconds is perpendicular to the initial velocity. Find, in either order, the value of $u$ and the value of $\sin \theta$. [5]
7 A parachutist of mass $m$ kg opens his parachute when he is moving vertically downwards with a speed of $50 \text{ m s}^{-1}$. At time $t$ s after opening his parachute, he has fallen a distance $x\text{ m}$ from the point where he opened his parachute, and his speed is $v \text{ m s}^{-1}$. The forces acting on him are his weight and a resistive force of magnitude $mv\text{ N}$.
(a) Find an expression for $v$ in terms of $t$. [6](b) Find an expression for $x$ in terms of $t$.
(c) Find the distance that the parachutist has fallen, since opening his parachute, when his speed is $15 \text{ m/s}^{-1}$.Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. ...
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