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A particle executes the motion described by x(t) = x_{0} (1 - e^{-\gamma t}) where t \geq 0, x0 > 0

a) Where does the particles start and with what velocity?

b) 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.

Answers (1)

Here, x(t)=x_{0}[1-e^{-\gamma t}]

So, v(t)=\frac{dx(t)}{dt}

                =\frac{d}{dt}[x_{0}(1-e^{-\gamma t})]

                =+x_{0}\gamma e^{-\gamma t}\; \; \; \; \; \; ..........(i)

&

a(t)=\frac{dv}{dt}

 =-x_{0}\gamma ^{2}e^{-\gamma t}\; \; \; \; \; \; ..........(ii)

(i) x(0) = x_{0} [1 - e^{0}]

                = x_{0} (1 - 1) =0

v(0) = x_{0}\gamma e^{0}

           =x_{0}\gamma

Thus, x=0 is the starting point of the particle and its velocity is v_{0}=x_{0}\gamma

(b) x (t)\ is,

Maximum at t = \infty        since t = \infty [x(t)]_{max} = \infty

Minimum at t = 0           since at t = 0,[x(t)]min = 0

v(t) is,

maximum at t = 0           since t=0, v(0) = x_{0}\gamma

minimum at  t = \infty         since, t = \infty, v(\infty ) = 0

a(t) is,

maximum at t = \infty        since at t = \infty, a(\infty )=0

minimum at t = 0            since at t = 0, a(0)=-x_{0}\gamma ^{2}

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