A bicycle racer sprints near the end of a race to clinch a victory. The racer ha

Question

A bicycle racer sprints near the end of a race to clinch a victory. The racer has an initial velocity of 108 m/s and accelerates at the rate of 0.55 m/s^2 for 7.25 s to accomplish this. (a) What is his final velocity, in meters per second, after this period of acceleration? (b) The racer continues at this velocity to the finish line if he was 345 m from the finish lane when he started to accelerate, how much less time did it take him to finish then if had had not accelerated? (c) One other racer was 7.5 m ahead when the winner started to accelerate, but was too tired to speed up and traveled at 11.6 m s until the finish line. If the winner continues to cycle at the same speed after crossing the finish line (to celebrate his victory), how far ahead of the loser will the winner be, in meters, when the loser finishes?

Explanation / Answer

a)

Vi = initial velocity = 10.8 m/s

a = acceleration = 0.55

t = time = 7.25 s

Vf = final velocity

using the equation

Vf = Vi + at

Vf = 10.8 + (0.55) (7.25) = 14.8 m/s

b)

X = total displacement = 345 m

t = time of travel during acceleration = 7.25 sec

X1 = displacement during acceleration

using the equation

X1 = Vi t + (0.5) a t2

X1 = (10.8) (7.25) + (0.5) (0.55) (7.25)2

X1 = 92.75 m

X' = distance travelled at constant speed = X - X1 = 345 - 92.75 = 252.25 sec

t' = time taken to travel at constant speed = X'/Vf = 252.25/14.8 = 17.04 sec

total time taken when racer accelerates

tacc = t + t' = 7.25 + 17.04 = 24.3 sec

tconst = time taken when racer doesn't accelerate = X/Vi = 345/10.8 = 31.94 sec

time saved = tconst - tacc = 31.94 - 24.3 = 7.64 sec

c)

Xloser = distance travelled by loser = 345 - 7.5 = 337.5 m

Vloser = constant speed of loser = 11.6 m/s

time taken by the loser to cross the finish line = tloser = Xloser /Vloser = 337.5/11.6 = 29.1 sec

twinner = time taken by winner = 31.94 sec

d = distance by which the winner is ahead = Vf (twinner - tloser) = 14.8 (31.94 - 29.1) = 42.03 m

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