A small rock sinking through a fluid experiences an exponentially decreasing acc

Question

A small rock sinking through a fluid experiences an exponentially decreasing acceleration as a function of time given by a(t) = g*e-bt , where b is a positive constant that depends on the shape and size of the rock and the physical properties of the water and g is the gravitational constant on earth. (a) What is the value of the acceleration at t = 0? (b) Based upon this result, derive an expression for the position of the rock as a function of time. Assume that its initial velocity and position is zero.

Explanation / Answer

given expression = a(t) = g*e - b*t

part(a)

at t=0

a = g*e - b*0

a = g*e

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part(b)

v(t) = integration a(t)*dt

v(t) = integration(g*e - bt)*dt

v(t) = g*e*t - b*t^2/2 + c1

given at t = 0 initial velocity v = 0

0 = g*e*0 + b*t + c1

c1 = 0

v(t) = g*e*t - b*t^2/2

again position x(t) = integration v(t)*dt

x(t) = integration(g*e*t - b*t^2/2 )*dt

x(t = g*e*t^2/2 - b*t^3/6 + c2

given at t = 0 positionx = 0

c2 = 0

x(t) = g*e*t^2/2 - b*t^3/6 <----------answer

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