数量关系之行程问题答题技巧
行程问题的重点在于三个量:路程、速度、时间,考来考去总是这三个点,那命题人如何增加难度呢?一是改变考查形式,比如直接求速度变成间接求解,二是增加因素,比如流水对船速的影响、车身长对路程的影响,等等。但归根究底还是考一个公式:路程=速度*时间,命题就围绕这个公式展开,一般都是已知一个或多个运动过程,每个运动过程包含三个量:路程、速度、时间,与此同时,不同的运动过程间这三个量必然存在某个共通点,比如路程相同,或者相同时间。因此,行程问题的基本解题思路就是:分析题干中的每一个运动过程,结合问题看未知量、找出已知量,如果有多个运动过程,找出彼此之间共通点,从一点延伸到面,列出数学表达式,思路一目了然。1、行程问题之相遇问题答题技巧
相遇问题是行程问题的一种考查形式,指两人(或两车等)从两地出发相向而行的行程问题,是研究“速度” 、“相遇时间”和“两地距离”三者之间的数量关系的应用题。三个量中比较难理解一点就是相遇时间,两人同时出发、同时到达某一点。很明显,运动时间相同,这个时间就称为“相遇时间”,做题时要谨记这个等量关系,是隐含的已知条件。尤其,近年来考题难度有所增加,单一的相遇问题很少考,综合题比较多,因此,做题时一定要思路清晰,抓准核心,当题中涉及相遇问题时,谨记“相遇时间相同”这一点,利用等量关系巧妙求解未知量,化未知为已知,结合其他已知条件解出最终答案。
2、行程问题之追击问题答题技巧
追及问题指的是两人(物)在行进过程中同向而行,快行者从后面追上慢行者的行程问题。它考虑的是两人(物)在相同时间内所行的路程差。命题人一般会从三个角度命题,直线运动中有两个:“同地不同时出发型”和“同时不同地出发型”;还有一个是环形运动中的“同时同地出发型”,这里要注意一点,它的路程差是一个隐含的已知条件,与追上次数有关。第一次追上,路程差是一个周长,第N次追上,路程差是n个周长,做题时如果不明白这一点,很难理清思路。这三类大家不仅要记得,还要学会辨别,如果是考追及问题,先理清它的类别,根据类别找准路程差,将其代入追及问题特有的公式“路程差=速度差*追及时间”,列出数学表达式,求解未知量。
但这只是基本的解题思路,现在的考题难度越来愈大,一道题可能涉及多个追及过程,两两相关,如果想正确解题, 一看你能否找准每一个“路程差”,二看你的火眼金睛跟思维清晰度。比如这样一道题。
甲、乙二人练习跑步,若甲让乙先跑10米,则甲跑5秒可追上乙,若乙比甲先跑2秒,则甲跑4秒能追上乙,则甲每秒跑多少米?( )
A 2 B 4 C 6 D 7
很明显,题中涉及两次追及,一一分析,第一次甲乙两人是同时不同地,找准路程差:10米,追及时间是5秒,思维清晰的话,根据这两个量很快就能想到公式“路程差=速度差*时间”,进而得到:速度差=10/5=2。下面看第二个追及过程,依然是同时不同地型,路程差就是乙2秒跑的路程,追及时间4秒,但注意不要忘记前面求出的速度差2,因此,路程差=4*2=8,即乙2秒跑了8米,速度=8/2=4,那么,甲的速度是4+2=6。答案是C。
3、行程问题之火车过桥问题答题技巧
火车过桥问题其实就是行程问题,重点在于三个量:路程、速度、时间,只是考查的运动物体不是人,而是火车,那为什么要专门拿出来研究呢?因为,火车不像人,人过桥,桥多长,人走的路程就多长;火车就不一样了,它的车身有一定的长度,如下图,车头A上桥后,只有当它到达A1后才能说火车过桥完毕,此时,火车上任选一个点,很明显可看出,其移动的距离即路程等于“桥长+车长”,并不单单只有桥长。因此,解题时千万不要忽略了“车长”这个因素。当然,还有很多跟火车类似的物体,也不要忽略了其本身的长度,比如一列队伍,队尾出桥才是过桥完毕,其路程也应该等于“桥长+队伍长度 ”。此外,也不一定是过桥,可能是过山洞,可能是经过某个人,无论如何,都要注意车长这个关键因素,而行程问题的基本公式“路程=速度*时间”也要跟着产生相应地变化:桥长+车长=速度*时间。
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