A particle moves along a straight line, and its position is given by s(t) = t^3 - 6t^2 + 9t meters. Determine the time at which the particle changes direction.
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To determine the time at which the particle changes direction, we need to find the values of t for which the particle's velocity changes sign.
The particle's velocity function is the derivative of the position function:
v(t) = s'(t) = 3t^2 - 12t + 9.
To find the time at which the particle changes direction, we need to find the zeros of the velocity function.
Setting v(t) = 0 and solving for t, we have:
0 = 3t^2 - 12t + 9.
Dividing through by 3, we get:
0 = t^2 - 4t + 3.
Factoring the expression on the right side, we have:
0 = (t - 1)(t - 3).
So, t = 1 or t = 3.
Therefore, the particle changes direction at t = 1 second and t = 3 seconds.
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