一名骑车者以 $a\,\mathrm{m\,s^{-2}}$ 的恒定加速度沿水平直路行驶。她经过点 $A$、$B$、$C$ 和 $D$。距离 $AB$ 和 $CD$ 分别为 $50\,\mathrm{m}$ 和 $100\,\mathrm{m}$。她从 $A$ 至 $B$、从 $B$ 至 $C$ 以及从 $C$ 至 $D$ 所需的时间均为 $10\,\mathrm{s}$。

求 $a$ 的值及骑车者从 $A$ 至 $D$ 的速率增加量，求解顺序不限。

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 $10\,\mathrm{m\,s^{-2}}$。

设A处速率u，第三个十秒从C开始时速率u+20a，写两段位移方程相减。

Let $u$ be the speed at $A$. For the first ten seconds,

At $C$, after 20 seconds, the speed is $u+20a$. For the next ten seconds,

Subtracting (1) from (2),

The journey from $A$ to $D$ lasts 30 seconds, so the increase in speed is

也可用匀加速运动一段的平均速度等于该段中点时刻的瞬时速度。

For constant acceleration, the average velocity over a time interval equals the velocity at its midpoint.

Taking $t=0$ at $A$, the velocity at $t=5$ is $50/10=5\,\mathrm{m\,s^{-1}}$, and the velocity at $t=25$ is $100/10=10\,\mathrm{m\,s^{-1}}$. Hence

Over the 30 seconds from $A$ to $D$, the speed increases by