motion in a plane
DESCRIPTION
Motion in a Plane. Chapter 3. Position Vector. Goes from the origin to the object – point p. Average Velocity Vector. Change in displacement over change in time. Instantaneous velocity vector. Instantaneous velocity. In two dimensions. Average acceleration vector. - PowerPoint PPT PresentationTRANSCRIPT
Physics Chapter 3 1
Motion in a Plane
Chapter 3
Physics Chapter 3 2
Position Vector
• Goes from the origin to the object – point p
x
y
rP
Physics Chapter 3 3
Average Velocity Vector
• Change in displacement over change in time
t
rvav
Physics Chapter 3 4
Instantaneous velocity vector
t
rv
t
0lim
Physics Chapter 3 5
Instantaneous velocity
• In two dimensions
22yx vvvv
x
y
v
vtan
Physics Chapter 3 6
Average acceleration vector
t
vaav
Physics Chapter 3 7
Instantaneous acceleration vector
t
va
t
0lim
Physics Chapter 3 8
Example
• A particle has x and y coordinates (4.0 m, 2.0 m) at time t1 = 2.0 s and coordinates (7.0 m, 6.0 m) at time t2 = 2.5 s. Find the components of the average velocity and the magnitude and direction of the average velocity during this time interval.
Physics Chapter 3 9
Projectile Motion
• Projectile – any body given an initial velocity which then follows a path (trajectory) based on gravitational acceleration and air resistance
• Thrown ball• Bullet• Dropped package
Physics Chapter 3 10
Projectile motion
• We will neglect air resistance• We will also neglect curvature and rotation
of the earth
Physics Chapter 3 11
Projectile Motion
• Consists of both horizontal and vertical motion
• We will break these problems into x and y components to make them easier to solve
Physics Chapter 3 12
Projectile motion
• Two dimensional – gravity only acts vertically
• We assign y as the vertical direction• We assign x as the horizontal direction
0xa gay
Physics Chapter 3 13
Projectile Motion
200 2
1tatvxx xx tavv xxx 0
xx vv 0 tvxx x 0
gtvv yy 0 200 2
1gttvyy y
0 0
Physics Chapter 3 14
Projectile Motion
• If we express initial velocity in terms of its magnitude and angle with the x-axis
000 cosvv x
000 sinvv y
Physics Chapter 3 15
Projectile Motion
• We can calculate the projectile’s speed and the direction of its velocity
22yx vvv
x
y
v
vtan
Physics Chapter 3 16
Trajectory shape
• Projectiles always travel in parabolas
Physics Chapter 3 17
Example
• A policeman chases a thief across city rooftops. They are both running at 5 m/s when they come to a gap between buildings that is 4 m wide and has a drop of 3 m.
• The thief leaps at 5 m/s at an angle of 45°. Does he clear the gap?
• The policeman leaps at 5 m/s horizontally. Does he clear the gap?
Physics Chapter 3 18
Example0x
0y
Physics Chapter 3 19
Example – the thief
tvxx x 0
tvx x
45 @ s
m 50 v
00 x
45coss
m 5
xv
tx
45cos
s
m 5
Physics Chapter 3 20
Example – the thief
200 2
1gttvyy y
20 2
1gttvy y
0
00 y
45 @ s
m 50 v 45sin
s
m 50
yv
Physics Chapter 3 21
Example – the thief
0m 345sins
m 5
s
m 9.4 2
2
tt
02
10
2 ytvgt y
m 3
roof, thehits hewhen
y
Physics Chapter 3 22
Quadratic formula
02 cbxax
a
acbbx
2
42
Physics Chapter 3 23
Example – the thief
0m 345sins
m 5
s
m 9.4 2
2
tt
2
2
2
sm
9.42
m 3sm
9.4445sinsm
545sinsm
5
t
Physics Chapter 3 24
Example – the thief
2
2
2
2
2
sm
8.9
sm
8.58sm
5.12sm
3.54
t
2sm
8.9
sm
8.44sm
3.54t
Physics Chapter 3 25
Example – the thief
22 sm
8.9
sm
9.4or
sm
8.9
sm
98.11tt
s 22.1t
Physics Chapter 3 26
Example – the thief
• Yes, he clears the gap
tx
45cos
s
m 5
s 22.145coss
m 5
x
m 32.4x
Physics Chapter 3 27
Example – the policeman
• Everything is the same, except that
0 @ s
m 50 v
s
m 5xv
00 yv
Physics Chapter 3 28
Example – the policeman
tx
s
m 5
g
yt
21
2
2
2
1gty
Physics Chapter 3 29
Example – the policeman
2
2
sm
9.4
m 3t
m 3
roof, thehits hewhen
y
22 s 61.0t
s 78.0t
Physics Chapter 3 30
Example – the policeman
• No, he does not clear the gap
tx
s
m 5
s 78.0s
m 5
x
m 9.3x
Physics Chapter 3 31
Example 2• A rookie quarterback throws a football
with an initial upward velocity component of 16.0 m/s and a horizontal component of 20.0 m/s.
a) How much time is required for the football to reach the highest point of the trajectory?
Physics Chapter 3 32
Example 2 a)
• At the highest point, vy must be zero.
• Before that, it was moving up, and after that it is moving down, so it must momentarily stop at that point.
gtvv yy 0
0yvs
m 0.160 yv
Physics Chapter 3 33
Example 2 a)
t
2s
m 8.9
s
m 0.160
2sm
8.9
sm
0.16t
s 6.1t
Physics Chapter 3 34
Example 2 b)
• b)How high is this point?
200 2
1gttvyy y
00 ys
m 0.160 yv
s 6.1t
Physics Chapter 3 35
Example 2 b)
2
2s 6.1
s
m 8.9
2
1s 6.1
s
m 16
y
m 5.12m 6.25 y
m 13.1y
Physics Chapter 3 36
Example 2 c)
• c) How much time is required for the ball to return to its original height?
200 2
1gttvyy y
00 y
s
m 0.160 yv
0y
Physics Chapter 3 37
Example 2 c)
20 2
10 gttv y
02
10
2 tvgt y
02
10
yvgtt
Physics Chapter 3 38
Example 2 c)
0t 02
1or 0
yvgt
yvgt 02
1
g
vt y02
Physics Chapter 3 39
Example 2 c)
2sm
8.9
sm
162
t
s 2.3t
Physics Chapter 3 40
Example 2 d)
• d) How does the answer to part c) compare to the answer to part a)?
• It is double. When something is thrown upward, it always takes the same amount of time to go up as it does to come down.
• So the total flight time (back to the original height) is always twice the time to the highest point.
Physics Chapter 3 41
Example 2 e)
• e) How far has the football traveled horizontally?
tvxx x 0
00 xs
m 20xv
s 2.3t
Physics Chapter 3 42
Example 2 e)
s 2.3s
m 20
x
m 64x
Physics Chapter 3 43
Normal acceleration
• Normal means perpendicular• The component of acceleration that is
perpendicular to the path• Shows change in direction of velocity
normaa or
Physics Chapter 3 44
Tangential acceleration
• Parallel to the path• Shows change in magnitude of velocity
(change in speed)
tanaa or
Physics Chapter 3 45
Uniform Circular Motion
• When an object moves around a circle with constant speed, the acceleration vector points towards the center of the circle.
• The velocity is tangent to the circle. In order for the speed to remain constant, the acceleration must be towards the center of the circle.
• We call this centripetal acceleration.
Physics Chapter 3 46
Uniform circular motion
• See Fig. 3-11 on page 63
R
s
v
v
1
sR
vv 1
tR
sv
t
vaav
1
t
s
R
va
t
0
lim
R
va
2
Physics Chapter 3 47
Uniform circular motion
• The period of any motion is the time it takes to make one complete cycle. It is denoted with the letter T.
• In this case it’s the time to go around the circle once.
t
dvav
T
Rv
2
2
24
T
Ra
Physics Chapter 3 48
Example
• The radius of the earth’s orbit around the sun is 1.49 x 1011 m, and the earth travels around the sun in 365.25 days.
• What is the earth’s orbital speed in m/s?• What is the acceleration of the earth
toward the sun in m/s2?