科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
1. Two uniform smooth spheres $A$ and $B$ of equal radii have masses $4m$ and $m$ respectively. Sphere $B$ is at rest on a smooth horizontal surface. Sphere $A$ is moving on the surface with speed $u$ and collides directly with sphere $B$. After the collision, the momentum of $A$ is three times the momentum of $B$. Find the value of the coefficient of restitution $e$. [4]
2 A particle $P$ of mass $m$ is moving in a horizontal circle with angular speed $\omega_1$ on the smooth inner surface of a hemispherical shell of radius $r$. The angle between the upward vertical and the normal reaction of the surface on $P$ is $\theta_1$, where $\tan \theta_1 = \frac{3}{4}$.
When the angular speed is increased to $\omega_2$, the angle between the upward vertical and the normal reaction of the surface on $P$ becomes $\theta_2$, where $\tan \theta_2 = \frac{4}{3}$.
Find the ratio $\frac{\omega_1}{\omega_2}$. [4]3 A particle $P$ is moving in a straight horizontal line. At time $t$s, the displacement of $P$ from a fixed point $O$ on the line is $x$m and the velocity of $P$ is $v$ms$^{-1}$. The acceleration of $P$ is $\frac{1}{2}(v^2 + 4)$ ms$^{-2}$ in the direction $PO$. Initially $P$ is at $O$ and is moving with velocity $2$ms$^{-1}$.
(a) Find an expression for $x$ in terms of $t$. [5](b) Find the time when $P$ next goes through $O$. [2]
4 One end of a light elastic string of natural length $a$ and modulus of elasticity $5mg$ is attached to a fixed point $O$. Two particles, $P$ and $Q$, of masses $m$ and $4m$ respectively are attached to the other end of the string and they hang vertically in equilibrium. Particle $Q$ is then detached from the string, hence releasing particle $P$ from rest.
Find, in terms of $a$, the length of the string when the speed of particle $P$ is first equal to $\sqrt{\frac{2}{5}}ag$. [6]7 © UCLES 2025 9231/33/O/N/25 [Turn over]
A uniform lamina $OABCD$ consists of a rectangle $OACD$ and a triangle $ABC$. The length of $OA$ is $ka$, the length of $OD$ is $2a$, the height of triangle $ABC$ is $h$ and angle $CAB$ is $45^\circ$ (see diagram). Relative to axes through $O$, parallel and perpendicular to $OA$ as shown, the centre of mass of triangle $ABC$ is $(\bar{x}, \bar{y})$.
(a) Show that $\bar{x}$ is $\frac{1}{3}(3ka + h)$, and find an expression for $\bar{y}$. [3]The lamina $OABCD$ is placed vertically on its edge $OA$ on a horizontal plane.
(b) Find, in terms of $a$ and $k$, the set of values of $h$ for which the lamina is in equilibrium. [4]
It is now given that $k = \frac{\sqrt{3}}{3}$ and that the lamina is on the point of toppling.
(c) Find, in terms of $a$, the coordinates of the centre of mass of the triangle $ABC$. [2]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 $O$. Initially $P$ is held with the string taut and making an angle of 60° with the upward vertical through $O$. The particle $P$ is projected perpendicular to the string in a downwards direction with speed $\sqrt{17ag}$. It then starts to move along a circular path in a vertical plane with centre $O$ (see diagram). At the lowest point of its path, vertically below $O$, the particle $P$ collides with a stationary particle $Q$.
(a) Find, in terms of $a$ and $g$, an expression for the speed of $P$ immediately before the collision with $Q$. [2]As a result of the collision, $P$ rebounds and moves back along a circular path with centre $O$. The string becomes slack when $P$ reaches the point on the circle vertically above $O$. (b) Find, in terms of $a$ and $g$, an expression for the speed of $P$ immediately after the collision with $Q$. The mass of particle $Q$ is $km$ and the collision between $P$ and $Q$ is perfectly elastic. (c) Find the value of $k$.
7 A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. The initial velocity of $P$ is $25\text{ms}^{-1}$ at an angle $\theta$ above the horizontal, where $\tan \theta = \frac{4}{3}$. At point $A$, the direction of motion of $P$ makes an angle of $45^\circ$ with the downward vertical through $A$.
(a) By differentiating the equation of the trajectory or otherwise, find the coordinates of $A$. [5]At point $A$, the particle strikes a fixed smooth barrier, rebounds, and lands on the horizontal plane. The barrier is inclined at an angle of 45° to the horizontal.
(b) Find the speed of $P$ immediately before it collides with the barrier. [3]
(c) Given that the coefficient of restitution between the barrier and the particle is $\frac{1}{9}$, find the horizontal distance travelled by $P$ after it strikes the barrier. [4]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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