A motorcyclist who is moving along an x-axis directed toward the east has an acc

Question

A motorcyclist who is moving along an x-axis directed toward the east has an acceleration given by a = a0(1 - t/A), where a0 is a positive constant with the units of m/s2, t is measured in seconds, and A is a positive constant with the units of seconds. At t = 0, the velocity and position of the cyclist are v0 and x0. In terms of the variables given above, (a) what is the maximum velocity achieved by the cyclist? (b) Find a formula for the position of the cyclist as a function of t.

(a) vmax=

(b) x =

Explanation / Answer

(a)
at time t, the speed of the motorcyclist is :

v(t) = v0 + a*t = v0 + a0 (1 - t/A)*t = v0 + a0 (t - t2 /A)

to find the max speed, consider:

v(t)' = dv(t)/dt = a0 (1-2t/A)

let v'(t) = 0, we have a0 (1-2t/A) = 0 , => t = A/2

so when time t = A/2, the speed is max

vmax = v0 + a0 (A/2 - (A/2)2 /A) = v0 + a0 (A/4)

b)

formula for the position of the cyclist as a function of t is:

x =x0 +   v0 * t + (1/2)a t2 = v0 * t + (1/2) (1 - t/A)* t2 = x0 + v0 * t + (1/2) (t2 - t3/A)

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