质点 $P$ 以速率 $u$ 在一个固定的光滑水平面上运动。质点与一个固定的光滑竖直挡板碰撞。碰撞前一瞬间，$P$ 的运动方向与挡板成 $\alpha$ 角，其中 $\tan\alpha=\frac{12}{5}$。

碰撞后 $P$ 的动能是碰撞前动能的 $\frac4{13}$ 倍。

(a) 求 $P$ 与挡板之间的恢复系数。

(b) 第二个光滑竖直挡板固定于该水平面及第一个挡板上。两个挡板之间的夹角为 $120^\circ$。质点 $P$ 随后与第二个挡板碰撞，碰撞后立即以速率 $w$ 弹回，运动方向与第二个挡板成 $\theta$ 角。恢复系数与第一次碰撞时相同。

求 $\theta$ 的值，保留 $3$ 位有效数字。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

Where a numerical value for the acceleration due to gravity ($g$) is needed, use $10\,\mathrm{m\,s^{-2}}$.

需要使用重力加速度 $g$ 的数值时，取 $10\,\mathrm{m\,s^{-2}}$。

平行于障碍物的分量不变，垂直分量反向并乘恢复系数 $e$。用两分量的平方和表示碰后动能比。

The given tangent gives $\cos\alpha=5/13$ and $\sin\alpha=12/13$. After impact, the parallel component is $5u/13$ and the normal component has magnitude $12eu/13$. The kinetic energy ratio therefore gives

Thus $25+144e^2=169(4/13)=52$, giving $144e^2=27$ and $e^2=3/16$. The non-negative coefficient is $\boxed{e=\sqrt3/4}$.

第一次反弹后，先求运动方向与第一面障碍物的夹角。两障碍物之间是 $120^\circ$，所以在第二面入射时，与它的锐角为 $60^\circ$ 减去前一个夹角。第二次碰撞把这个角的正切乘 $e$。

Let $\phi$ be the angle of the first rebound to the first barrier. Then

The path between the barriers and the two barrier segments form a triangle with angle $120^\circ$ at their meeting point. Thus the angle of approach to the second barrier is $\delta=60^\circ-\phi$. The compound-angle formula gives

At the second collision, the normal component is multiplied in magnitude by $e$ and the parallel component is unchanged. If $V$ is the speed between the barriers, then $w\sin\theta=eV\sin\delta$ and $w\cos\theta=V\cos\delta$. Dividing gives $\tan\theta=e\tan\delta=(\sqrt3/4)(\sqrt3/7)=3/28$, so $\boxed{\theta=6.12^\circ}$ to three significant figures.