Ch 03: 2D Motion (Projectile Motion)WorksheetSee all chapters
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Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
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Ch 20: The First Law of Thermodynamics
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Solution: A bird flies in a vertical 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

Problem

A bird flies in a vertical 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

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:

PVddtdtA

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

r(t)=v(t) dt+r0

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

a(t)=dv(t)dt

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

xn dt=1n+1xn+1  (for example, x2dt=13x3

ddt(xn)=nxn-1  (for example, ddtx3=3x2)

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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