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.

Answer 1

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