HMMT 二月 2005 · CALC 赛 · 第 7 题

HMMT February 2005 — CALC Round — Problem 7

Two ants, one starting at ( − 1 , 1), the other at (1 , 1), walk to the right along the 2 parabola y = x such that their midpoint moves along the line y = 1 with constant 1 speed 1. When the left ant first hits the line y = , what is its speed? 2

解析

Two ants, one starting at ( − 1 , 1), the other at (1 , 1), walk to the right along the 2 parabola y = x such that their midpoint moves along the line y = 1 with constant 1 speed 1. When the left ant first hits the line y = , what is its speed? 2 √ Solution: 3 3 − 3 1 3 When the left ant first hits the line y = , the right ant hits the line y = . The 2 2 √ √ 2 1 6 3 left ant is then at ( − , ), and the right ant is at ( , ). Let the left ant have 2 2 2 2 velocity with components v and v , the right ant velocity with components w and x y x √ √ v w dy y y w . Since = 2 x , = − 2 and = 6. Since the midpoint of the ants moves y dx v w x x 1 1 at speed 1 along the line y = 1, ( v + w ) = 1 and ( v + w ) = 0. Therefore, x x y y 2 2 √ √ √ √ √ 2 6 √ √ 2 v = − v = w = 6 w = 6(2 − v ). Solving for v gives = 3 − 3. Then x y y x x x 6+ 2 the speed of the left ant is √ √ √ √ √ 2 2 2 2 v + v = v + ( − 2 v ) = 3 | v | = 3 3 − 3 . x x x y x