Help would be greatly appreciated. A mad engineer chinstrap penguin creates a wa

Question

Help would be greatly appreciated.

A mad engineer chinstrap penguin creates a waterjet propulsion device that causes time-dependent acceleration a(t) = b exp(-ct) i, in the horizontal direction i over frictionless ice. The acceleration decays exponentially as the device runs out of fuel. The penguin straps this device to herself and turns it on from rest. Draw a picture and label all quantities. What is her velocity v(t) as a function of time? If b = 9.8m/s^2 is the value of her maximum acceleration, and c = 0.1s^-1 is the decay rate of the acceleration, what is the maximum velocity that the penguin can reach with her device?

Explanation / Answer

b) a(t) = bexp(-ct)

dv(t) /dt = b exp(-ct)

dv(t) = b exp(-ct) dt

integrating both sides

v(t) = (-b/c) exp(-ct) + C , C is an integration constant

she starts from rest at t=0, v(t) =0 , applying the initial condition in the above equation

C = b/c

Her velocity as a time of t

v(t) = (b/c) (1-exp(-ct) )

c) b = 9.8 m/s2 , c =0.1 /s

v(t) = 98(1-exp(-0.1t))

The maximum value of v will occur when the exp. term becomes 0 , at infinity or approx. 50 s

i.e. max V = 98 m/s

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