10. A cyclist races in a straight line with a speed function in meters per secon

Question

10. A cyclist races in a straight line with a speed function in meters per second given by where a,b and c are constants. Define a function in GeoGebra of this form and adjust the constants so that after a long time the cyclist is going 13 m/s, and early on reaches a maximum speed of 20 m/s, 8 seconds from the start. What values of a,b and c accomplish this (with correct units)? How far will the cyclist have gone in 60s? What will the cyclist's speed be at t-10s? How long will t take to cover one ilometer? Please use GeoGebra to do this. At what time is the acceleration maximum? Minimum

Explanation / Answer

from the given data
speed vx = at^2*e^(-bt) + c
here a, b and c are constants

now, vmax = 20 m/s at t = 8s
now,
d(vx)/dt = at^2*e^(-bt)*(-b) + 2at*e^(-bt)
at vmax, d(vx)/dt = 0
hence
64e^(-8b)*(b) = 2t*e^(-8b)
b = 1/4 per s

also, v(t-> inf) = 13 m/s

now, lim t-> inf (t^2/e^(bt)) = lim t-> inf (2t/be^(bt)) = lim t-> inf (2/b^2e^(bt)) = 0
hence
c = 13 m/s

also,
vmax = 20
20 = a(64)*e^(-0.25*8) + 13
a = 0.808178 m/s^3

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