HMMT 二月 1998 · CALC 赛 · 第 6 题

英文解析

6. Problem: Edward, the author of this test, had to escape from prison to work in the grading roomtoday. He stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow.
   The guard fired an arrow with an initial velocity of 100 ft/s. At the same time, Edward started running
   2 2
   away with an acceleration of 1 ft/ s . Assuming that air resistance causes the arrow to decelerate at 1 ft/ sand that it does hit Edward, how fast was the arrow moving at the moment of impact (in ft/s)?
   1 2
   Solution: We use the formula for distance, d = at + vt + d . Then after t seconds, Edward is at
   1 2 1 220
   location 1875 + (1)( t ) from the prison. After t seconds, the arrow is at location ( − 1)( t ) + 100 t from the
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   prison. When the arrow hits Edward, both objects are at the same distance away from the tower. Hence
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   2 2 2
   1875 + (1)( t ) = ( − 1)( t ) + 100 t . Solving for t yields t − 100 t + 1875 = 0 ⇒ t = 25 or t = 75. Then it
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   must be t = 25, because after the arrow hits Edward, he will stop running.
   After 25 seconds, the arrow is moving at a velocity of 100-25(1) = 75 ft/s .