A particle executes the motion described by x(t) = x0(1 − e−γt); t ≥ 0x0 > 0.x(t) = x0(1 − e−γt); t ≥ 0x0 > 0..
(a) Where does the particle 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.
x(t) = x0(1 − e−γt)
When t = 0; x(t) = x0(1−e−0) = x0(1−1) = 0.
Step 2: Calculate the velocity, v(t) = dxdt
v(t) = dx(t)dt = x0γe−γt.
Step 3: Calculate initial velocity by putting t = 0
V(0) = x0γe0 = x0γ.
(b)Hint: If f is increasing dfdt > 0, if decreasing dfdt < 0.
Step 1: Check weather f is increasing or decreasing
For x(t)
As here dxdt = x0γe−γt > 0
i.e., x is increasing
For velocity, v(t)
dv(t)dt = -x0γ2e-γt < 0
i.e., v(t) is decreasing
For acceleration, a(t)
da(t)dt = x0γ3e-γt >0
i.e., a(t) is increasing
Step 2: Calculate maximum and minimum values
For increasing, the maximum value is at t = ∞ and minimum value is at t = 0,
For decreasing, the maximum value is at t = 0 and the minimum value is at t = ∞.
For x(t)
As it is increasing
So,
x(t) is maximum when t=∞; x(t)]max=x0x(t) is minimum when t=0; x(t)]min=0v(t) is decreasingSo,v(t) is maximum when t=0; v(0)=x0γv(t) is minimum when t=∞; v(∞)=0a(t) is increasing So,a(t) is maximum when t=∞; a(∞)=0a(t) is minimum when t=0; a(0)=−x0γ2
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