📝 题目
3.4.10 某建筑工程打地基时,需要汽锤将桩打进土层,汽锤每次击打,都将克服土层对桩的阻力而作功. 设土层对桩的阻力的大小与桩被打进地下的深度成正比 (比例系数为 $k\left( {k > 0}\right)$ ),汽锤第一次击打将桩打进地下 $a\mathrm{\;m}$ . 根据设计方案,要求汽锤每次击打桩时所作的功与前一次击打时所作的功之比为常数 $r$ $\left( {0 < r < 1}\right)$ . 向
(1)汽锤击打桩 3 次后,可将桩打进地下多深？
(2)若击打次数不限,汽锤至多能将桩打进地下多深？
💡 答案与解析
3.4.10 (1) 设第 $n$ 次击打后,桩被打进地下 ${x}_{n}$ ,第 $n$ 次击打时,汽锤所做的功为 ${w}_{n}\left( {n = 1,2,\cdots }\right)$ . 依题设,当桩被打进地下的深度为 $x$ 时,土层对桩的阻力为 ${kx}$ ,所以
$$ {W}_{1} = {\int }_{0}^{{x}_{1}}{kx}\mathrm{\;d}x = \frac{1}{2}k{x}_{1}^{2} = \frac{1}{2}k{a}^{2}, $$
$$ {W}_{2} = {\int }_{{x}_{1}}^{{x}_{2}}{kx}\mathrm{\;d}x = \frac{1}{2}k\left( {{x}_{2}^{2} - {x}_{1}^{2}}\right) = \frac{1}{2}k\left( {{x}_{2}^{2} - {a}^{2}}\right) . $$
又 ${W}_{2} = r{W}_{1} \Rightarrow {x}_{2}^{2} - {a}^{2} = r{a}^{2}$ ,即 ${x}_{2}^{2} = \left( {1 + r}\right) {a}^{2}$ . 进一步,因此有
$$ {W}_{3} = {\int }_{{x}_{2}}^{{x}_{3}}{kx}\mathrm{\;d}x = \frac{1}{2}k\left( {{x}_{3}^{2} - {x}_{2}^{2}}\right) = \frac{1}{2}k\left\lbrack {{x}_{3}^{2} - \left( {1 + r}\right) {a}^{2}}\right\rbrack . $$
再由 ${W}_{3} = r{W}_{2} \Rightarrow {r}^{2}{W}_{1} \Rightarrow {x}_{3}^{2} - \left( {1 + r}\right) {a}^{2} = {r}^{2}{a}^{2} \Rightarrow {x}_{3} = \sqrt{1 + r + {r}^{2}}a$ . 即汽锤击打桩 3 次后,可将桩打进地下 $\sqrt{1 + r + {r}^{2}}a\mathrm{\;m}$ .
(2)用数学归纳法,设 ${x}_{n} = \sqrt{1 + r + \cdots + {r}^{n - 1}}a$ ,则
$$ {W}_{n + 1} = {\int }_{{x}_{n}}^{{x}_{n + 1}}{kx}\mathrm{\;d}x = \frac{1}{2}k\left( {{x}_{n + 1}^{2} - {x}_{n}^{2}}\right) $$
$$ = \frac{1}{2}k\left\lbrack {{x}_{n + 1}^{2} - \left( {1 + r + \cdots + {r}^{n - 1}}\right) {a}^{2}}\right\rbrack . $$
由于
$$ {W}_{n + 1} = r{W}_{n} = {r}^{2}{W}_{n - 1} = \cdots = {r}^{n}{W}_{1} = \frac{1}{2}k{r}^{n}{a}^{2}, $$
故有
$$ {x}_{n + 1}^{2} - \left( {1 + r + \cdots + {r}^{n - 1}}\right) {a}^{2} = {r}^{n}{a}^{2} \Rightarrow $$
$$ {x}_{n + 1} = \sqrt{1 + r + \cdots + {r}^{n}}a = \sqrt{\frac{1 - {r}^{n + 1}}{1 - r}}a, $$
于是
$$ \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n + 1} = \sqrt{\frac{1}{1 - r}}a. $$
即若击打次数不限,汽锤至多能将桩打进地下 $\sqrt{\frac{1}{1 - r}}a\mathrm{\;m}$ .