1. Motion in Two and Three Dimensions
AP Physics Chapter 4
Motion in Two and Three Dimensions

2. AP Physics
Turn in Chapter 3 Homework, Worksheet, & Lab
Lecture
Q&A

3. 1-D Review Position: x, (m)
The 3 Great Equations of constant acceleration motion:
Displacement:
x, (m)
Velocity:
v, (m/s)
Acceleration:
a, (m/s2)

4. Position 1D: x  3D: r or r = x i +y j + z k or
x: scalar component of r in i direction
y: scalar component of r in j direction
z: scalar component of r in k direction

5. Position is a 3D vector k z r  y j  x i

6. Displacement r = r2 – r1  rf – ri
r = (xf – xi)i + (yf – yi)j + (zf – zi)k or

7. Velocity
Instantaneous velocity in the x direction depends on the displacement only in the x direction.
The instantaneous velocity v of a particle is always tangent to the path of the particle.

8. Instantaneous acceleration in the x direction depends on the change of velocity only in the x direction.
Acceleration

9. Huh?
Most often, we find the derivative to find acceleration when the position is given as an equation, but then it is more like a pure math problem.

10. Example: Pg77-7 An ion’s position vector is initially r = 5. 0i – 6
Example: Pg77-7 An ion’s position vector is initially r = 5.0i – 6.0j + 2.0k, and 10 s later it is r = -2.0i + 8.0j –2.0k, all in meters. In unit-vector notation, what is its vavg during the 10 s?
Given: ri = 5.0i – 6.0j + 2.0k, rf = -2.0i + 8.0j –2.0k, t = 10s,
vavg = ?

11. Practice: Pg77-11 The position r of a particle moving in an xy plane is given by r = (2.00t3 – 5.00t)i + (6.00 – 7.00t4)j, with r in meters and t in seconds. In unit-vector notation, calculate (a) r, (b) v, and (c) a for t = 2.00 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle’s path a t = 2.00 s?

12. Practice: Pg77-11 The position r of a particle moving in an xy plane is given by r = (2.00t3 – 5.00t)i + (6.00 – 7.00t4)j, with r in meters and t in seconds. In unit-vector notation, calculate (a) r, (b) v, and (c) a for t = 2.00 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle’s path a t = 2.00 s? x y • d)
Velocity is tangent to the path, so what is asked for is the angle between v and +x axis.

13. Projectile Motion: 2-D
For projectile motion, choose coordinates so that the motion is confined in one plane
 2D motion x y v0
v0y
v0x
Initial velocity 0

14. Package Dropped From Airplane (Snapshot every second)

15. Snapshots Horizontal Direction: velocity is _______ constant
Vertical Direction: velocity is __________
increasing

16. Ball projected straight up from truck (Snapshot every second)

17. Snapshots

18. Projectile Motion: Horizontal and Vertical Displacements

19. Velocity Components of Projectile

20. Independence of Motion
From observations and experiments:
The horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.
Connection:
Both horizontal and vertical motions are functions of time. Time connects the two independent motions.
Though independent, these two motions are related to each other by time.

21. Projectile Motion Breakdown
Horizontal: constant velocity
Vertical: Constant acceleration (ay = g, downward)
ay = g if downward is defined as +y direction
ay = -g if upward is defined as +y direction

22. Example: A small ball rolls horizontally off the edge of a tabletop that is 1.2 m high. It strikes the floor at a point 1.52 m horizontally from the table edge. a) How long is the ball in the air? b) What is its speed at the instant it leaves the table?
Define coordinates as to the right.
v0x = ?, y = 1.2m, x = 1.52m, x0 = 0, y0 = 0,
vx = const, ay = g = 9.8m/s2, v0y = 0 x y v0
a) Constant acceleration in y direction.
y = 1.2m
x = 1.52m
b) Constant velocity in x direction.

23. Practice: 78-30 You throw a ball with a speed of 25
Practice: You throw a ball with a speed of 25.0 m/s at an angle of 40.0o above the horizontal directly toward a wall. The wall is 22.0 m from the release point of the ball. A) How long does the ball take to reach the wall? B) How far above the release point does the ball hit the wall? C) What are the horizontal and vertical components of its velocity as it hits the wall? D) When it hits, has it passed the highest point on its trajectory?
Set up the coordinates as to the left. Then
vx=const, ay=-g, vox=v0cos40.0o=19.2m/s,
voy = v0sin40.0o = 16.1m/s,x0=0, y0=0,
x=22.0m y
a) t = ? v0
Constant v in horizontal direction.
40o x

24. Practice: 78-30 You throw a ball with a speed of 25
Practice: You throw a ball with a speed of 25.0 m/s at an angle of 40.0o above the horizontal directly toward a wall. The wall is 22.0 m from the release point of the ball. A) How long does the ball take to reach the wall? B) How far above the release point does the ball hit the wall? C) What are the horizontal and vertical components of its velocity as it hits the wall? D) When it hits, has it passed the highest point on its trajectory?
b) y = ?
Constant acceleration in y direction.
c) vx = ? and vy = ?
d) It has not yet passed the highest point on its trajectory since it is still going up because vy = 4.8 m/s > 0.

25. Highest Point of Trajectory
Trajectory: path of projectile
Minimum speed at top, but
vy = 0
vx = v0x = v0 cos 0
Maximum Height is

26. Symmetry of Trajectory
Upward motion and downward motion are symmetric at same height.
Upward total time is equal to downward total time if landing point is at same height as initial point.
At the same height, speed is the same but at different directions.
vx stays unchanged.
vy remains unchanged at the same magnitude but changes in direction.
vy is up when object is on its way up.
vy is downward when object is on its way down.

27. Symmetry of Trajectory (2)v vy vx vx vy v
At the same height, the speed is the same.
It takes the same amount of time to go up as to fall down.

28. Equation of Path No time involved.
Can be used to find x or y when the other is given.
Parabolic equation  Trajectory is parabolic.
Valid only when upward is defined as the +y direction as origin is set at the initial point.

29. Horizontal Range R is a distance and, therefore, a scalar.
Independent of frame of reference. (Same result would arrive if different frames of references were chosen.)
Two angles (complementary) with the same initial speed give the same range.
Horizontal range is maximum when the launch angle is 45o.
Valid only when the landing point and initial point are at the same height.

30. Example:82-77 A projectile is fired with an initial speed vo = 30
Example:82-77 A projectile is fired with an initial speed vo = 30.0 m/s from level ground at a target on the ground, at distance R = 20.0 m, as shown in Fig What are the (a) least and (b) greatest launch angles that will allow the projectile to hit the target?

31. Set up the coordinates as in the diagram. a)
Practice: During a tennis match, a player serves the ball at 23.6 m/s, with the center of the ball leaving the racquet horizontally 2.37 m above the court surface. The net is 12 m away and 0.90 m high. When the ball reaches the net, (a) does the ball clear it and (b) what is the distance between the center of the ball and the top of the net? Suppose that, instead, the ball is served as before but now it leaves the racquet at 5.00o below the horizontal. When the ball reaches the net, (c) does the ball clear it and (d) what now is the distance between the center of the ball and the top of the net?
Set up the coordinates as in the diagram. a) y
0 = 0, vox = vo = 23.6m/s, voy = 0,
ax = 0, ay = -g, x0 = 0, y0 = 2.37m,
x = 12m, y = ? 0
Constant velocity in x direction: x
Constant acceleration in y direction:
 Yes

32. Practice: 78-26 During a tennis match, a player serves the ball at 23
Practice: During a tennis match, a player serves the ball at 23.6 m/s, with the center of the ball leaving the racquet horizontally 2.37 m above the court surface. The net is 12 m away and 0.90 m high. When the ball reaches the net, (a) does the ball clear it and (b) what is the distance between the center of the ball and the top of the net? Suppose that, instead, the ball is served as before but now it leaves the racquet at 5.00o below the horizontal. When the ball reaches the net, (c) does the ball clear it and (d) what now is the distance between the center of the ball and the top of the net? c)
o = -5.00o, voy = vosin o = 23.6m/s·sin(-5.00o)= -2.06m/s,
vox = vocos o=23.5m/s, ax = 0, ay = -g, xo = 0, yo = 2.37m,x = 12m,
y = ?
Constant velocity in x-direction:
Constant acceleration in y-direction:
The ball will not clear the net now.

33. Decomposing motion More than one way to decompose a vector. vx
More than one way to decompose a vector. y v
Fall at same time x
x: constant v
y: constant a,  v y
x: constant v
y: constant a, 

34. Uniform Circular Motion
Circular path or circular arc
Uniform = Constant speed (Constant velocity? why?) v
Magnitude of velocity:
constant
Direction of velocity: v
changing
Velocity:
changing

35. Centripetal Acceleration
v: speed of particle
r: radius of circle or circular arc
, where v
Direction of acceleration is always toward the center of circle (or circular arc)
 Centripetal a a v a v
for uniform circular motion at any time.

36. Direction of Acceleration
a in the same direction as v:
 Speed:
increases
Tangential Acceleration
a opposite to v:
Change:
speed
 Speed:
decreases
a  v:
 Speed:
does not change
Radial Acceleration
 Direction of velocity:
changing
Change:
direction of velocity

37. Direction of Acceleration (2)
Tangential acceleration changes only the speed.
Radial acceleration changes only direction of velocity.
Both tangential and radial components of acceleration are needed to change both the direction and magnitude of a velocity.
Radial direction
Tangential direction

38. Uniform Circular Motion
Period: Time for one complete cycle
Frequency: Number of cycles per unit time
Unit:

39. Example: A) What is the centripetal acceleration of an object on the Earth’s equator owing to the rotation of the Earth? B) What would the period of rotation of the Earth have to be for objects on the equator to have a centripetal acceleration equal to 9.8 m/s2?

40. Practice: 79-48 A rotating fan completes 1200 revolutions every minute
Practice: A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of 0.15 m. (a) Through what distance does the tip move in one revolution? What are (b) the tip’s speed and (c) the magnitude of its acceleration? (d) What is the period of the motion?

41. Relative Motion vAB: velocity of A with respect of B,
velocity of A relative to B,
velocity of A measured by B, or
velocity of A in frame of reference B.
No real cancellation, just helps us to remember. 1D
2D,
or 3D?

42. Inertial Frame of Reference
Inertial Frame of Reference: Frame that stays at rest or moves at constant velocity
Law of Physics:
aPA = aPB
aPA: Acceleration of particle P as measured by observer A (or in Frame A).
Observers on different inertial frames of references will measure the same acceleration for a moving particle.

43. Example: A cameraman on a pickup truck is traveling westward at 20 km/h while he videotapes a cheetah that is moving westward 30 km/h faster than the truck. Suddenly, the cheetah stops, turns, and then runs at 45 km/h eastward, as measured by a suddenly nervous crew member who stands alongside the cheetah’s path. The change in the animal’s velocity took 2.0 s. What are the (a)magnitude and (b) direction of the animal’s acceleration according to the cameraman and (c) magnitude and (d) direction according to the nervous crew member?
T: Truck (or cameraman), G: ground (or nervous crew member), C: cheetah
Let eastward be the positive direction. Then
vTG = -20km/h,
vCT.i = -30km/h,
vCG.f = 45km/h,
t = 2.0s
b) East
d) East

44. Upstream has been defined as + direction.
Practice: A boat is traveling upstream in the positive direction of an x axis at 14 km/h with respect to the water of a river. The water is flowing at 9.0 km/h with respect to the ground. What are the (a) magnitude and (b) direction of the boat’s velocity with respect to the ground? A child on the boat walks from front to rear at 6.0 km/h with respect to the boat. What are the (c) magnitude and (d) direction of the child’s velocity with respect to the ground?
Upstream has been defined as + direction.
b) upstream
d) downstream C W B +x

45. Practice: 80-59 Snow is falling vertically at a constant speed of 8
Practice: Snow is falling vertically at a constant speed of 8.0 m/s. At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of 50 m/s?
Set up frame of reference, then
vsg
We have
vdg x y
Then we can find    or