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

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.

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.

Motion in 2D & 3D With Calc

Motion in 2D & 3D With Calc

Motion in 2D & 3D With Calc

Motion in 2D & 3D With Calc