Question

A particle moves along the x-axis so that at any time t greater than 0, its position is given by x(t) = t^3 - 3t^2 - 9t. For what values of t is the particle at rest?

Answer 1

The particle is at rest when t = 3 and t = -1.

Step-by-step explanation:

To determine when the particle is at rest, we need to find when the velocity of the particle is equal to zero. The velocity of the particle is the derivative of its position with respect to time. So, we need to find when the derivative of the position function, x(t), is equal to zero.

Let's calculate the derivative of x(t):

x'(t) = 3t^2 - 6t - 9

Now, we set x'(t) equal to zero and solve for t:

3t^2 - 6t - 9 = 0

Using the quadratic formula, we can find the values of t:

t = (-(-6) ± √((-6)^2 - 4(3)(-9))) / (2(3))

Simplifying the expression inside the square root:

t = (6 ± √(36 + 108)) / 6

t = (6 ± √144) / 6

t = (6 ± 12) / 6

t = (6 + 12) / 6 or t = (6 - 12) / 6

t = 18 / 6 or t = -6 / 6

t = 3 or t = -1

Therefore, the particle is at rest at t = 3 and t = -1.

Learn more about Particle motion