Instantaneous Acceleration in 2D Video Lessons

Concept

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.

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?