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Question
A particle executes the motion described by x(t) = x0(1 − e −γt); t ≥ 0, x0 > 0. Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time.
Solution
Given, `x(t) = x_0 (1 - e^(-γt))`
`v(t) = (dx(t))/(dt) = x_0 γe^(-γt)`
`a(t) = (dv(t))/(dt) = - x_0 γ^2 e^(-γt)`
x(t) is maximum when t = ∞ [x(t)]max = x0
x(t) is minimum when t = 0 [x(t)]min = 0
v(t) is maximum when t = 0; v(0) = x0γ
v(t) is minimum when t = ∞; v(∞) = 0
a(t) is maximum when t = ∞; a(∞) = 0
a(t) is minimum when t = 0; a(∞) = − x0γ2
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