What happens to the other mouse when one of two mice that ran in opposite directions has covered a certain distance? - briefly
The other mouse has traveled the same distance, but in the opposite direction, so the separation between them equals twice that distance.
What happens to the other mouse when one of two mice that ran in opposite directions has covered a certain distance? - in detail
When two rodents start from the same point and move in opposite directions, the distance each travels is governed by their individual speeds and the elapsed time. If the first animal reaches a predefined length d, the second animal will have covered a distance equal to its speed multiplied by the same time interval.
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Identical speeds – If both mice run at the same constant velocity v, the time required for the leading mouse to achieve distance d is t = d ⁄ v. During this interval the opposite‑moving mouse also travels d, ending up at a point symmetric to the first, separated by 2 d.
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Different speeds – Let the leading mouse move with speed v₁ and the opposite mouse with speed v₂. The time for the former to reach distance d remains t = d ⁄ v₁. The latter’s displacement becomes v₂·t = v₂·d ⁄ v₁. Consequently, the separation between the two equals d + (v₂·d ⁄ v₁).
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Variable speed – If either mouse accelerates or decelerates, the distance covered by the second mouse is obtained by integrating its instantaneous velocity over the time required for the first mouse to travel d. The resulting position reflects the cumulative effect of the speed profile.
If the forward‑moving mouse stops precisely after covering distance d, the opposite mouse continues according to its current velocity. Its subsequent position after any additional interval Δt is given by the previous displacement plus v₂·Δt.
In all cases, the spatial relationship between the two rodents remains determinable from the known distance of the first and the velocity characteristics of the second. The outcome for the remaining mouse is therefore a predictable function of time and speed, without ambiguity.