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# Problem: An electron’s position is given by  r(t) = 3.00t  î − 4.00t 2  ĵ + 2.00  k̂, with t in seconds and  r  in meters.(a) In unit-vector notation, what is the electron’s velocity  v(t)?(b) At t = 2.00 s, what is  v  in unit-vector notation?(c) At t = 2.00 s, what is the magnitude of  v?(d) At t = 2.00 s, what is the angle of  v relative to the positive direction of the  x-axis?

###### FREE Expert Solution

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:

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

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

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

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!

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

$\overline{)\mathbf{|}\stackrel{\mathbf{⇀}}{\mathbit{v}}\mathbf{|}{\mathbf{=}}\sqrt{{{\mathbit{v}}_{\mathbf{x}}}^{\mathbf{2}}\mathbf{+}{{\mathbit{v}}_{\mathbit{y}}}^{\mathbf{2}}}}$

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

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

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###### Problem Details

An electron’s position is given by  r(t) = 3.00t   − 4.00t 2   + 2.00  , with t in seconds and  r  in meters.
(a) In unit-vector notation, what is the electron’s velocity  v(t)?
(b) At t = 2.00 s, what is  v  in unit-vector notation?
(c) At t = 2.00 s, what is the magnitude of  v?
(d) At t = 2.00 s, what is the angle of  v relative to the positive direction of the  x-axis?

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.