Problem: Suppose the position of an object is given by r = (3.0t2 i − 6.0t3 j) m .(a) Determine its velocity v and acceleration a, as a function of time,(b) Determine r and v at time t = 2.5 s.

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This problem requires us to determine velocity and acceleration as a function of time and to determine and v at a given time.

Anytime we're given a position, velocity, or acceleration function and asked to find one or more of the others, we know it's a motion problem with calculus. A PVA diagram like the one below can help remind you of the relationships between the three functions:

$\mathbit{P}\begin{array}{c}{\mathbf{←}}\\ {\mathbf{\to }}\end{array}\underset{\frac{\mathbit{d}}{\mathbit{d}\mathbit{t}}}{\overset{{\mathbf{\int }}{\mathbit{d}}{\mathbit{t}}}{\mathbit{V}}}\begin{array}{c}{\mathbf{←}}\\ {\mathbf{\to }}\end{array}\mathbit{A}$

To get from position to velocity, we differentiate the position function.

$\overline{)\stackrel{\mathbf{⇀}}{\mathbit{v}}{\mathbf{\left(}}{\mathbit{t}}{\mathbf{\right)}}{\mathbf{=}}\frac{\mathbit{d}\stackrel{\mathbf{⇀}}{\mathbit{r}}\mathbf{\left(}\mathbit{t}\mathbf{\right)}}{\mathbit{d}\mathbit{t}}}$

Also, to get acceleration, we differentiate the velocity function:

$\overline{)\stackrel{\mathbf{⇀}}{\mathbf{a}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{\right)}}{\mathbf{=}}\frac{\mathbf{d}\stackrel{\mathbf{⇀}}{\mathbf{v}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}}{\mathbf{d}\mathbf{t}}}$

Remember the power rule of differentiation.

$\overline{)\frac{\mathbit{d}}{\mathbit{d}\mathbit{t}}\mathbf{\left(}{\mathbit{t}}^{\mathbit{n}}\mathbf{\right)}{\mathbf{=}}{\mathbit{n}}{{\mathbit{t}}}^{\mathbit{n}\mathbf{-}\mathbf{1}}}$  (for example, $\frac{\mathbit{d}}{\mathbit{d}\mathbit{t}}{\mathbit{t}}^{\mathbf{3}}\mathbf{=}\mathbf{3}{\mathbit{t}}^{\mathbf{2}}$)

Whenever you take the derivative of a vector, make sure to do the operation on each component (î, ĵ, and k̂) separately—they're independent of each other and shouldn't get mixed up!

(a) We're asked to determine velocity and acceleration as functions of time.

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Problem Details

Suppose the position of an object is given by r = (3.0t2 i − 6.0t3 j) m .
(a) Determine its velocity v and acceleration a, as a function of time,
(b) Determine r and v at time t = 2.5 s.