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# 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?

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 Mirones' class at MDC.