1. Ch 4: Motion in Two and Three Dimensions
Fundamentals of Physics Halliday, Resnick, Walker
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8/24/2012

2. Introduction
This chapter combines the concepts of 1-D motion and vectors to describe motion in two and three dimensions.
What are some examples of 2-D and 3-D motion?
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8/24/2012

3. Agenda Position and Displacement Velocity (Average and Instantaneous)
Acceleration (Average and Instantaneous)
Projectile Motion
Uniform Circular Motion
Relative Motion in 1-D and 2-D
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4. Vocabulary Projectile Motion Uniform Circular Motion
Centripetal Acceleration
Period of Revolution
Relative Motion
Reference Frame
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5. Position Vector, r(t)
The location of an object in space is specified by a position vector, r.
The position, r(t), is a function of time.
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8/24/2012

6. Rabbit Example: Position r(t)
A ‘wascally’ rabbit’s position is given by the parametric equations
r(t) = xî + yĵ where
x(t) = −0.31t t + 28
y(t) = 0.22t2 − 9.1t + 30
Find r at t = 15 s.
x(15) = 66m, y(15) = −57m
|r| = 87 m, θ = −41o
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8/24/2012

7. Displacement Vector, Δr
Displacement , Δr, of an object is the difference between two positions.
Δr = r(t2) – r(t1) = r2 – r1
= (x2 − x1) î + (y2 − y1) ĵ
= Δx î + Δy ĵ
Displacement is not necessarily the same as the distance traveled. Why?
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8/24/2012

8. Rabbit Example: Plot r(t)
Use parametric mode on graphing calculator to plot the rabbit’s position.
Note: This is y vs x, not x vs t.
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8/24/2012

9. Rabbit Example: Displacement Δr
Determine the rabbit’s displacement Δr over the interval t = 10s to t = 20s.
r(10) = 69î − 39ĵ m
r(20) = 48î − 64ĵ m
Δr = r(20) − r(10)
= (48 − 69) î + (−64 −(−39)) ĵ
= −21 î −25 ĵ m
|r| = 33 θ = 230o
***Sketch the vectors***
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8/24/2012

10. Average Velocity Vector, vavg
vavg = displacement / time (same as 1-D case)
vavg is in the same direction of Δr
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8/24/2012

11. Rabbit Example: vavg(t)
Determine the rabbit’s average velocity over the interval t = 10 s to t = 20 s
Determine vavg
vavg = −2.1 î −2.5 ĵ m/s
vavg = o
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8/24/2012

12. Instantaneous Velocity, v(t)
v(t) is the velocity of the object at an instant in time.
v(t) is the instantaneous rate of change of the object’s position, r, with respect to time, t.
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8/24/2012

13. Speed, |v(t)|
Speed is the magnitude (or absolute value) of the instantaneous velocity.
|v| = (vx2 + vy2 + vz2) ½
Speed is always ≥ 0.
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8/24/2012

14. Direction of the Instantaneous Velocity, v(t)
The direction of v is always tangent to the object’s path at the object’s position.
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8/24/2012

15. Rabbit Example: Velocity, v
v is tangent to the rabbit’s path at any instant.
Determine v at t = 15 s.
vx(t) = dx/dt = −0.62t + 7.2
vy(t) = dy/dt = 0.44t − 9.1
v (15) = −2.1î − 2.5ĵ m/s
= 3.3 −130o
= o
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16. Average Acceleration, aavg
Average acceleration, aavg, is the ratio of the object’s change in velocity, Δv, and the corresponding time interval, Δt .
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17. Instantaneous Acceleration, a(t)
a(t) is the acceleration of the object at an instant in time.
a(t) is the instantaneous rate of change of v with respect to t.
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18. Rabbit Example: Acceleration, a
Determine a at t = 15 seconds.
ax(t) = dvx/dt = −0.62 m/s2
ay(t) = dvy/dt = 0.44 m/s2
a (15) = −0.62î − 0.44ĵ m/s2
= o
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8/24/2012

19. Speeding up or Slowing down?
If an object is changing speed and / or changing direction, then it is accelerating.
Given that θ is the angle between a and v when they are tail to tail, then
If θ = 0o, the object is speeding up only.
If θ < 90o, the object is speeding up and changing direction.
If θ = 90o, the object’s speed is constant, but it’s changing direction.
If θ > 90o, the object is slowing down and changing direction.
If θ = 180o, the object is slowing down only.
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8/24/2012

20. Projectile Motion Special case of 2-D motion
Horizontal motion: ax = 0 so vx = constant
Vertical motion: ay = g = constant so the constant acceleration equations apply.
Assumptions:
Horizontal and vertical motions are independent of each other
Air resistance (i.e., drag) can be ignored.
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21. Projectile Motion Illustrated
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22. Projectile Motion Equations
Trajectory Equation: objects follow a parabolic path.
Range Equation: gives range R = Δx only when Δy = 0.
What angle gives maximum range?
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23. Motion with Constant Acceleration
v = vo + at
x − xo = vot + ½ at2
v2 = vo2 + 2a(x − xo)
x − xo = ½ (vo + v)t
x − xo = vt − ½ at2
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8/24/2012

24. Free-Fall Acceleration Equations
If +y is vertically up, then the free-fall acceleration due to gravity near Earth’s surface is a = − g = − 9.8 m/s2.
v = vo − gt
y − yo = vot − ½ gt2
v2 = vo2 − 2g(y − yo)
y − yo = ½ (vo + v)t
y − yo = vt + ½ gt2
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25. Projectile Motion: Monkey-Hunter Demo
The projectile and can fall at the same rate.
The point of intersection depends on the launch velocity angle and speed.
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26. Problem Solving Techniques
Event 1
Event 2
Time, t
x position
x velocity
x accel.
Determine the two events that define the start and end of the motion.
Identify the known and unknown quantities for the x and y motions.
The x and y equations are independent, but are linked by time t.
Event 1
Event 2
Time, t
y position
y velocity
y accel.
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27. Projectile Motion: Rescue Problem
vo = 198 km/h, horizontal
h = 500 m
At what line of sight angle φ should the pilot release the raft to reach the swimmer?
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28. Projectile Motion: Pirate Problem
Range of pirate ship is R = 560 m.
The cannon ball’s initial speed is vo = 82 m/s.
At what angle(s) should the fort aim the cannon to hit the pirate ship?
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29. Uniform Circular Motion
Another special case of 2-D motion.
An object travels in a circle of radius r at constant speed v.
Centripetal (“center seeking”) acceleration
Direction:
perpendicular to v
directed towards center of circle
Magnitude: a = v2/r
What causes centripetal acceleration?
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30. Period of Revolution
Period of revolution = Time required for the object to go around the circle once.
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31. UCM: Top Gun Problem
A Test Pilot flying a F-22 fighter aircraft at a speed of 2500 km/h initiates a horizontal turn of radius 5.80 km.
What is the pilot’s centripetal acceleration?
Given that loss of consciousness occurs at approximately 5g of acceleration, is the pilot in danger? Explain.
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32. Relative Motion in 1-D
An observer watches two joggers running by at 6 km/hr. What is the relative speed between the two joggers?
Observers measure position, velocity, and acceleration relative to their frame of reference.
The reference frame is the physical object to which we attach our coordinate system.
Examples: ground, car, plane, ship
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33. Relative Motion in 1-D Illustrated
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34. Relative Motion Notation
Observers in reference frames A and B measure the position of the same object P.
xPA = xPB + xBA
Position of Frame B relative to Frame A
Position of P measured
by observer in Frame A
Position of P measured
by observer in Frame B
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35. Relative Motion Equations, 1-D
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36. Relative Motion Equations, 2-D
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37. Relative Motion in 2-D Illustrated
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38. Frames of Reference Moving at Constant Relative Velocity
Observers in different frames of reference may measure different positions and velocities of an object.
The observers will measure the same acceleration if they move at constant velocity relative to each other.
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39. 1-D Frame of Reference Example
An observer on a Train throws Ball in x-direction with velocity of vBT.
Velocity of train relative to ground is vTG.
An observer on the ground measures velocity of ball, vBG.
vBG = vBT + vTG
vBG
km/h
vBT
vTG 50
100
−50
−25 25
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8/24/2012

40. 2-D Frame of Reference Example: Airplane Navigation
Givens:
Airspeed of plane is 215 km/h.
Wind velocity is 65.0 km/h directed 20o East of North.
The plane flies due East relative to the ground.
Determine the plane’s heading and ground speed.
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8/24/2012

41. Summary
2-D and 3-D motion combines the concepts of 1-D motion and vectors.
Know the various r, v, and a definitions.
Special cases of 2-D motion:
Projectile Motion
Uniform Circular Motion
Relative Motion and Frames of Reference.
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8/24/2012