Question

A particle moves along the x-axis so that at any time t≥0 its position is given by x(t) = t3 − 3t2 − 9t. For what values of t is the particle at rest?

Answer 1

To determine when the particle is at rest, we need to find the values of t such that the velocity of the particle is equal to zero. The velocity of the particle can be found by taking the derivative of the position function, x(t). So, we differentiate x(t) with respect to t to find the velocity function, v(t). Solving the equation 3t^2 - 6t - 9 = 0, we find that t = 3.96 s.

Step-by-step explanation:

To determine when the particle is at rest, we need to find the values of t such that the velocity of the particle is equal to zero. The velocity of the particle can be found by taking the derivative of the position function, x(t). So, we differentiate x(t) with respect to t to find the velocity function, v(t).

Starting with x(t) = t^3 - 3t^2 - 9t, we can differentiate it as follows:

v(t) = dx/dt = 3t^2 - 6t - 9.

To find when the particle is at rest, we need to find the values of t that make v(t) = 0:

0 = 3t^2 - 6t - 9.

Solving this quadratic equation, we find that t = 3.96 s.