Answer:
[tex](-12)[/tex] meters per second.
Explanation:
The velocity of a moving object is the rate of change in position. Since position is given as a function of time [tex]S(t) = t^{3} - 6\, t^{2} + 6[/tex], velocity of the object would be the first derivative of that function with respect to time: [tex]S^{\prime}(t)[/tex].
To find the velocity at the particular moment [tex]t = 2[/tex], apply the following steps:
To find the first derivative of [tex]S(t) = t^{3} - 6\, t^{2} + 6[/tex], apply the power rule of differentiation:
[tex]\begin{aligned}S^{\prime}(t) &= \frac{d}{d t}\left[t^{3} - 6\, t^{2} + 6\right] \\ &= 3\, t^{2} - (6)\, (2\, t) + 0 \\ &= 3\, t^{2} - 12\, t\end{aligned}[/tex].
In other words, velocity at time [tex]t[/tex] would be [tex]S^{\prime}(t) = (3\, t^{2} - 12\, t)[/tex].
Substitute in [tex]t = 2[/tex] and evaluate to find the velocity at time [tex]t = 2[/tex]:
[tex]S^{\prime}(2) = 3\, (2)^{2} - 12\, (2) = (-12)[/tex].
In this question, it is given that both position and time are in standard units, Since the expression for velocity is obtained from differentiating position with respect to time, that expression would also be in standard units: meter per second. Therefore, the velocity of this object at time [tex]t = 2[/tex] would be [tex](-12)[/tex] meters per second.