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Motion in 2D & 3D With Calc

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Sections
Intro to 2D Motion
Projectile Motion
More Projectile Motion
Initial Velocity in Projectile Motion
Circular Motion

Solution: A bird flies in the xy-plane with a velocity vector given by v = (2.4 − 1.6t2) 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 i

Problem

A bird flies in the xy-plane with a velocity vector given by v = (2.4 − 1.6t2) + 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?

Solution

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:

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

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:

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

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

$\overline{){\int }{{x}}^{{n}}{d}{t}{=}\frac{1}{n+1}{{x}}^{n+1}}$  (for example, $\int {x}^{2}dt=\frac{1}{3}{x}^{3}$

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

(a) The first part of the problem asks us to calculate the position vector as a function of time. Since the problem states that the bird starts at the origin at t = 0, the initial position ${\stackrel{⇀}{r}}_{0}=\stackrel{⇀}{0}$.

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