Instantaneous Acceleration in 2D Video Lessons

Concept

# Problem: A bird flies in a vertical  xy-plane with a velocity vector given by  v = (2.4 − 1.6t 2)  î + 4.0t  ĵ, where  v is in m/s and t is in seconds. The positive  y-direction is vertically upward. At  t t= 0 the bird is at the origin.(a) Calculate the position vector of the bird as a function of time.(b) Calculate the acceleration vector of the bird as a function of time.(c) What is the bird's altitude (y-coordinate) as it flies over  x = 0 for the first time after  t = 0?

###### FREE Expert Solution

We are asked to find the position, acceleration, and a specific coordinate for a bird in flight, given its velocity as a 2D function of time and initial position.

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 velocity to position, we integrate the velocity function. Make sure to add the initial position as a constant of integration!

To get from velocity to acceleration, we take the derivative of the velocity function:

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

We'll often use the power rule of integration and the power rule of derivation, which are:

(for example, $\mathbf{\int }{\mathbit{x}}^{\mathbf{2}}\mathbit{d}\mathbit{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}{\mathbit{x}}^{\mathbf{3}}$

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

Whenever you take the derivative or integral 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!

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###### Problem Details

A bird flies in a vertical  xy-plane with a velocity vector given by  v = (2.4 − 1.6t 2)   + 4.0t  , where  v is in m/s and t is in seconds. The positive  y-direction is vertically upward. At  t = 0 the bird is at the origin.
(a) Calculate the position vector of the bird as a function of time.
(b) Calculate the acceleration vector of the bird as a function of time.
(c) What is the bird's altitude (y-coordinate) as it flies over  x = 0 for the first time after  t = 0?