The displacement of a particle is given by: x=t^3-7/2t^2+2t-1. Find the particles initial displacement. And find when the particle is at rest.

initially, t=0, so x = -1

particle is at rest when dx/dt = 0
dx/dt = 3t^2 - 7t + 2
dx/dt=0 when t = 1/3 or 2

To find the particle's initial displacement, we need to find its position at time t=0. Simply substitute t=0 into the equation x=t^3-7/2t^2+2t-1:

x(0) = (0)^3 - 7/2(0)^2 + 2(0) - 1
= 0 - 0 + 0 - 1
= -1

So, the particle's initial displacement is -1 units.

To find when the particle is at rest, we need to find the time(s) when the particle's velocity (rate of change of displacement) is zero. Velocity can be obtained by taking the derivative of the displacement equation with respect to time.

First, let's find the velocity function by differentiating x(t) with respect to t:

v(t) = d/dt (t^3-7/2t^2+2t-1)
= 3t^2 - 7t + 2

Now, we can find the time(s) when the particle is at rest by solving the equation v(t) = 0:

3t^2 - 7t + 2 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, the quadratic factors nicely:

(3t - 1)(t - 2) = 0

Setting each factor equal to zero:

3t - 1 = 0 or t - 2 = 0

Solving for t:

3t = 1 or t = 2

t = 1/3 or t = 2

So, the particle is at rest at t = 1/3 and t = 2.