This problem requires us to determine **velocity** and **acceleration **as a function of *time* and to determine **r **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:

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

To get from __ position to velocity__, we

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

Also, to ** get acceleration**, we

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

Remember the power rule of differentiation.

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

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.

Suppose the position of an object is given by **r** = (3.0*t*^{2} **i** − 6.0*t*^{3} **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.

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Instantaneous Acceleration in 2D concept. You can view video lessons to learn Instantaneous Acceleration in 2D. Or if you need more Instantaneous Acceleration in 2D practice, you can also practice Instantaneous Acceleration in 2D practice problems.

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Our expert Physics tutor, Patrick took 3 minutes and 40 seconds to solve this problem. You can follow their steps in the video explanation above.

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Based on our data, we think this problem is relevant for Professor Efthimiou's class at UCF.