You are at a race track with special rules. The race is over a distance L, but y
ID: 1329742 • Letter: Y
Question
You are at a race track with special rules. The race is over a distance L, but you must begin at rest at x=0 and end at rest at x = L. Your car has maximum positive acceleration, a, and maximum deceleration, 2a. L is short enough that you do not reach your car's maximum speed. To cover the distance in the minimum time, tmin, for what fraction of that time should you accelerate and how much of the distance have you covered in that time? Assume that the acceleration and deceleration are constant, and assume that the transition from acceleration to deceleration is instantaneous.
Explanation / Answer
Let the accelerating time be t1 and decelerating time be t2
t1 + t2 = t
the maximum velocity that the car reaches in acceleration be v
=> for accelerating phase
From, v = u + a * t
v = a * t1 .......I
=>for decelerating phase
0 = v - 2a * t2
=>v = 2 * a * t2 .............II
from I and II t1 = 2t2 .........III
from III and t1 + t2 = t
we get t1 = 2t/3; t2 = t/3
This means car has to accelerate 2/3 rds of the time
distance travelled in accelerating phase is s = 1/2 * a * t1^2
=1/2 * a * 4/9 * t^2
= 2 / 9 a t^2
If you want the distance in terms of L, consider the distance travelled in decelerating phase also.
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