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
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
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Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

Solution: 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: 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:

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:

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

ddtxn=nxn-1  (for example, ddtx3=3x2)

(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 r0=0.

r(t)=v(t) dt+r0=2.4+1.6t2 dt i^+4.0t dt j^+r0=(2.4t+815t3) i^+(2.0t2) j^+0

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