This problem requires us to determine a **velocity** in unit-vector notation, its **magnitude**, and its **direction**, given an object's position as a 3D function of 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

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

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!

Also recall the equations to find a vector's magnitude and direction from its components:

$\overline{)\mathbf{\left|}\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}\mathbf{\right|}{\mathbf{=}}\sqrt{{{\mathit{v}}_{\mathbf{x}}}^{\mathbf{2}}\mathbf{+}{{\mathit{v}}_{\mathit{y}}}^{\mathbf{2}}}}$

$\overline{){\mathbf{tan}}{\mathit{\theta}}{\mathbf{=}}\frac{{\mathit{v}}_{\mathit{y}}}{{\mathit{v}}_{\mathit{x}}}}$

(a) We're asked to determine** velocity function** of the electron in unit-vector notation.

An electron’s position is given by **r**(*t*) = 3.00*t ***î** − 4.00*t ^{ }*

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.

What professor is this problem relevant for?

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

What textbook is this problem found in?

Our data indicates that this problem or a close variation was asked in Fundamentals of Physics - Halliday Calc 10th Edition. You can also practice Fundamentals of Physics - Halliday Calc 10th Edition practice problems.