橡皮筋问题 不知道以前有没有出过

新用户

一根可以无限均匀拉长的橡皮筋，初始长度为1米。有一个虫子以速度为０．１米／秒的速度从橡皮筋的一段开始往前爬。假如橡皮筋同时以每秒0.5米的速度拉申。

问：虫子能不能爬到橡皮筋的另一端？

假设虫子的寿命为无限长。www.ddhw.com

 

Math Genius

suppose at time t, the bug is at position x (relative to the original rope, i.e. x=0 is the beginning and x=1 is the end of the rope). Then we have, the bug's absolute position is:www.ddhw.com                                 (1+0.5t)*x,        and the bug's absolute speed is 0.1+0.5x,   so we have d((1+0.5t)*x)/dt = 0.1 +0.5x, the solution for x=1 is t=(e^5-1)*2, or something like that. I do not guarantee the accuracy of the final answer since it was 20 years ago when i solve my last differential equation       www.ddhw.com

huxlnn

答案是可以到达。基本分析是因为橡皮绳均匀拉伸，虫子不断往前，虫子背后的绳子长度越来越大，均匀拉伸所占的比率也越来越大，于是同时前面绳子的所占的拉伸比率越来越少。具体分析计算用到了微积分，虫子与前方的绳子端点的距离随着时间的增加而逐渐减少。 www.ddhw.com

野 菜 花

Let an be the original length of the part of the original rubber band the insect  crawls in the nth secend. The total length of the rubber band string after the nth second is 1+n*0.5 an:1=0.1:(1+n*0.5) an=0.1/(1+n*0.5)=1/(10+5n) So,  a1=1/15, a2=1/20, a3=1/25... a1+a2+...=1/5*(1/3+1/4+1/5+...) is infinite, since 1+1/2+1/3+...=infinite Therefore, finally the insect is able to reach the other end.    www.ddhw.com

新用户

  能否请给出分析计算过程？我没有答案

dky

  可以。