Question

Particle Motion A particle moves along a line so that its position at any time \(t \geq 0\) is given by the function \(s(t)=\) \(t^{3}-6 t^{2}+8 t+2\) where \(s\) is measured in meters and \(t\) is measured in seconds. (a) Find the instantaneous velocity at any time t. (b) Find the acceleration of the particle at any time t. (c) When is the particle at rest? (d) Describe the motion of the particle. At what values of t does the particle change directions?

Short answer

The velocity function is \(v(t) = 3t^{2}-12t+8\). The acceleration function is \(a(t) = 6t-12\). The particle is at rest at the times \(t\) where the velocity function is zero. The particle changes direction at the instances where the sign of the velocity changes. The motion of the particle can be described by examining the sign of the velocity and acceleration functions.

Step-by-step solution

Find the velocity as the derivative of the position function

Using power rule for derivatives, the derivative of \(t^{3}-6 t^{2}+8 t+2\) is \(3t^{2}-12 t+8\). This gives us the velocity function \(v(t)\).

Find the acceleration as the derivative of the velocity function

The derivative of \(3t^{2}-12 t+8\) is \(6t-12\). So, the acceleration function \(a(t)\) is \(6t-12\).

Find when the particle is at rest

We set the velocity function \(v(t) = 3t^{2}-12 t+8\) equal to zero and solve the quadratic equation. The solutions to this equation will give us the times \(t\) when the particle is at rest.

Determine when the particle changes direction

At the values of \(t\) where \(v(t) = 0\), we need to check the sign of velocities just before and just after these points. A change of sign implies a change in direction.

Describe the motion of the particle

With the obtained information, we can infer that if the velocity at \(t\) is positive, the particle is moving to the right while if it's negative, the particle is moving to the left. Furthermore, if the acceleration at \(t\) is positive, the particle is speeding up, and if it's negative, it's slowing down. The change of directions, points of rest and its motions can be described comprehensively by examining these aspects together.