甲乙丙相遇问题

Question

甲、乙两人从相距490米的A、B两地同时步行出发，相向而行。丙与甲同时从A地出发，在甲、乙二人之间来回跑步（遇到乙立即返回，遇到甲也立即返回）。已知丙每分钟跑240米，甲每分钟走40米，当丙第一次折返回来并与甲相遇时，甲、乙二人相距210米，那么乙每分钟走多少米？甲下一次遇到丙时，甲、乙相距多少米？

Thought

Analyzing movements

I'm thinking through the scenario where甲 walks towards乙,乙 heads towards甲, and丙 runs back and forth between them, all starting at different times.

Tracking movements

OK, let me see. C starts at A, heads towards B at 240 m/min, meets乙, turns back, and then meets甲 again, with甲 and乙 210 m apart.

Tracking progress

C heads towards乙, turns around to meet甲, and calculates the distance between甲 and乙, which is 210 m. Next, we'll define t1 as the time when C first meets乙.

Calculating meeting point

I’m calculating where C and乙 intersect by adding their distances. This gives me t1 = 490 / (240 + v), where 490 meters is the initial separation between A and B.

Charting C's return

C heads back to A, covering 240 t2 meters in t2 minutes. Simultaneously,甲 walks further from A, covering 40(t1 + t2) meters.

Aligning positions

Calculating that C and甲 meet at t2 = (5/7) t1, with甲 walking 40(t1 + t2) meters and乙 covering v(t1 + t2) meters, their total distance should be 280 meters.

Cracking the equation

By integrating previous findings, I substitute t1, rationalizing it, and streamline to find the ratio of (40 + v) to (240 + v).

Piecing it together

I determined乙's speed is 60 m/min. Now, C runs back towards乙, meeting甲. Next, I calculate the distance between甲 and乙 when C meets甲 again.

Mapping out positions

I’m working through the positions of C and甲 by adding up their distances. This shows their positions are aligned, with C running towards乙.

Tracking progress

C heads towards乙 at 240 m/min, while甲 walks towards B at 40 m/min. I’m calculating their positions to pinpoint when C meets甲 again.

Calculating meeting point

I’m working through determining the meeting point of C and甲 by calculating乙's position over time, ensuring the distance between甲 and乙 is consistently 210 meters at each step.

Verifying calculations

I’m checking the consistency of calculations for speed and meeting times. Ensuring the distance between甲 and乙 is 210 meters each time helps confirm these results.

Tracking movements

C runs towards乙 at 240 m/min,乙 heads towards A at 60 m/min, and甲 walks to B at 40 m/min. I’m figuring out their meeting points along the way.

Tracking their journey

C heads towards乙,甲 walks towards B, and乙 moves to A. The distance between甲 and乙 is consistently 210 meters. This consistency highlights their paths and speeds.

Calculating movement dynamics

C heads towards乙 at 240 m/min, while乙 walks towards A at 60 m/min. Meanwhile,甲 heads towards B at 40 m/min. I’m finding their meeting point.

Charting new directions

C meets乙 at 280 m from A, and I’m noting their speeds and positions. Then, C heads towards甲, who has walked 140 m from A. Next, C meets甲.

Calculating meeting points

I found that C meets甲 0.5 minutes after leaving乙, ensuring all positions and speeds align accurately. This calculation helps in understanding their relative movements effectively.

Confirming positions

I’ve checked C and甲's meeting at 160 m and calculated甲 and乙's positions at 4.0 minutes. They are 90 m apart, confirming their relative distances and movements.

Thought for 22 seconds

Answer

我们需要解决两个部分的问题：

乙的速度
甲下一次遇到丙时，甲与乙的距离

1. 乙的速度

已知条件：

A、B 两地相距 490 米。
甲的速度为 40 米/分钟。
丙的速度为 240 米/分钟。

步骤：

第一次相遇时：
- 设丙在时间 $t_1$ 分钟后第一次遇到乙。
- 在这段时间内，丙走了 $240t_1$ 米，甲走了 $40t_1$ 米，乙走了 $vt_1$ 米（ $v$ 为乙的速度）。
- 因为他们从相对方向出发，满足： $240t_1 + vt_1 = 490 \quad \Rightarrow \quad t_1 = \frac{490}{240 + v}$
丙返回并与甲相遇：
- 设从第一次相遇到与甲相遇的时间为 $t_2$ 分钟。
- 在这段时间内，丙返回并与甲相遇时： $240t_2 = 40(t_1 + t_2) \quad \Rightarrow \quad t_2 = \frac{5}{7}t_1$
- 在相遇时，甲与乙的距离为 210 米： $(40 + v)(t_1 + t_2) = 280$ 代入 $t_2 = \frac{5}{7}t_1$ ： $(40 + v)\left(\frac{12}{7}t_1\right) = 280 \quad \Rightarrow \quad \frac{(40 + v) \cdot 12}{7} \cdot \frac{490}{240 + v} = 280$ 解得 $v = 60$ 米/分钟。

2. 甲下一次遇到丙时，甲与乙的距离