1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall...
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Transcript of 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall...
![Page 1: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/1.jpg)
1
Topics
• Distance, Location, Speed
• Speed and Direction
• Directional quantities
• Acceleration
• Free Fall
• Graphs of Motion
• Derivatives and Integrals
![Page 2: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/2.jpg)
2
Average Speed
• distance: total path length
• speed: rate of travel (e.g. 50 mph)
• Average Speed: distance/time (e.g. 100m in 3.0s)
[m/s] timetravel
distances
![Page 3: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/3.jpg)
3
Displacement: Change in Position
if xxx SI Unit: meters (m)
![Page 4: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/4.jpg)
4
Velocity (m/s)
0 : : velocityaverage
tt
xvavg
0 : : velocityousinstantane
tt
xv
![Page 5: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/5.jpg)
5
Velocity Examples
• average velocity: 60mph toward Dallas
• instantaneous velocity: 11:47am: Northbound, 83mph
![Page 6: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/6.jpg)
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Example: Average Velocity
to = 0.0s, xo = 5.0m, vo = +2.0m/s
t = 1.2s, x = 3.08m, v = -5.2m/s
smss
mm
t
xvavg /6.1
0.02.1
00.508.3
Note that velocities always have directional information. Here the “-” sign means –x direction.
![Page 7: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/7.jpg)
7
Scalars & Vectors
• Scalar: size only
• e.g. speed, distance, time
• Vector: magnitude and direction
• e.g. displacement, velocity, acceleration
![Page 8: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/8.jpg)
8
A honeybee travels 2 km round trip before returning. Is the displacement for the trip the same as the distance traveled?
1 2
79%
21%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45
1. Yes
2. No
![Page 9: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/9.jpg)
9
Acceleration (m/s/s)
0 : :onaccelerati average
tt
vaavg
0 : :onaccelerati ousinstantane
tt
va
![Page 10: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/10.jpg)
10
Example: Car goes from 10m/s to 15m/s in a time of 2.0 seconds. Calculate the average acceleration.
m/s/s 5.20.0s-2.0
10m/s-15
t
vaavg
![Page 11: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/11.jpg)
11
Previous Example:
to = 0.0s, xo = 5.0m, vo = +2.0m/s
t = 1.2s, x = 3.08m, v = -5.2m/s
m/s/s 0.60.0s-1.2
2.0m/s-5.2-
t
vaavg
![Page 12: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/12.jpg)
12
Motion Diagrams
• velocity arrow and position• zero velocity is a “dot”• acceleration & net-force directions: parallel to v• Example: slowing, reversing direction
![Page 13: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/13.jpg)
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Kinematic Equations of Constant Acceleration
atvv o :velocity
tvvx o )( : velocityaverage 21
221 :ntdisplaceme attvx o
xavv o 2 :squared-v 22
![Page 14: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/14.jpg)
14
Displacement and x vs. t Graph
![Page 15: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/15.jpg)
15
x vs. t Graph
• slope is velocity
![Page 16: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/16.jpg)
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v vs. t Graph• slope is acceleration
atvv o
![Page 17: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/17.jpg)
17
Human Acceleration
mat 20221
In the 1988 Olympics, Carl Lewis reached the 20m mark in 2.96s. Calculate average acceleration.
20)96.2( 221 a
ssms
ma //56.4
)96.2(
2022
![Page 18: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/18.jpg)
18
Cheetah Acceleration
A cheetah can accelerate from 0 to 20m/s in 2.0s. What is the average acceleration?
ssms
sm
t
vva o //10
0.2
/)020(
![Page 19: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/19.jpg)
19
Ex: V2 EquationApproximate Stopping Accelerations in m/s/s:
Dry Road: ~ 9 (anti-lock) ~ 7 (skidding)
Wet Road: ~ 4 (anti-lock) ~ 2 (skidding)
At 60mph = 27m/s, what is the stopping distance of a skid on a wet road?
feet) 006(about 182
)2(2270
222
22
mx
x
xavv o
![Page 20: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/20.jpg)
20
Free-Fall
• only gravity acts
• air-friction is negligible
• a = 9.8m/s/s downward
![Page 21: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/21.jpg)
21
Calculus of Linear Motion
• derivatives and integrals
• Examples:
• dx/dt = v dv/dt = a
• d/dt(3 + 4t + 5t2) = 4 + 10t
• v = integral of acceleration
![Page 22: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/22.jpg)
22
Velocity
a(t)dtvv o
22545 ttdtt
Example:
![Page 23: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/23.jpg)
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Summary:• speed: rate of travel• average speed: distance/time.• displacement: change in position• velocity: rate position changes• acceleration: rate velocity changes• kinematic equation set• free fall: constant acceleration.• graphs and slopes• derivatives and integrals of polynomials
![Page 24: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/24.jpg)
24
![Page 25: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/25.jpg)
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Example: A solid metal ball is projected directly upward with velocity +5.0m/s. How high does it go? How long does it take to return to same height?
mh
gh
gh
yavv o
28.1
6.19/25)2/(25
250
222
22
sgt
gt
t
gttgtt
attvy o
02.18.9/10/10
0)5(
0
)5(50
21
212
21
221
![Page 26: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/26.jpg)
26
Case Study: 100 meter track-race
1. a = const., 0-60 m 2. top speed of 16 m/s at 60 m. 3. a = 0, 60-100 m
velocity vs time
0.002.004.006.008.00
10.0012.0014.0016.0018.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
t(s)
velo
city
(m/s
)
![Page 27: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/27.jpg)
27
st
t
t
at
attvx
t
o
5.78/60
860
60
060216
21
221
221
2/13.25.7/16
16
016
sma
ta
at
atvv o
a) Acceleration and Time
100m Race
![Page 28: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/28.jpg)
28
st
t
attvx o
5.216/40
01640
221
b) Time and Distance: Last 40meters of race at constant speed of 16m/s.
Race Time = tI + tII = 7.5s + 2.5s = 10.0s
100m Race
![Page 29: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/29.jpg)
29
v = vo + at.16 = 0 + a(7.5)a = 16/7.5 = 2.13 m/s2.
c) We can also use time found in part (a) in velocity equation to get the acceleration of the runner in 1st part of the race.
x = vavgt = {(vo + v)/2}t = {(0 + 16)/2)}(7.5) = (8)(7.5) = 60m.
d) Distance using vavg
![Page 30: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/30.jpg)
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Position vs time
0.00
20.00
40.00
60.00
80.00
100.00
120.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
t(s)
po
sit
ion
(m)
![Page 31: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/31.jpg)
31
Example: An object has velocity of +2.0m/s at x = 5.0m and at t = 0.0s. At t = 1.2s it has velocity of -5.2m/s and position x = 3.08m.
Average Acceleration:
ssmss
smsm
t
vaavg //0.6
0.02.1
/0.2/2.5
smsssmsmatvv o /2.5)2.1)(//0.6(/0.2
Using v(t) equation:
Consistent answer:
How long did it take the object to reach v = 0?
sssm
sm
a
vt
atv
o
o
33.0//0.6
/0.200
0
![Page 32: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/32.jpg)
32
![Page 33: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/33.jpg)
33
A train moves along a straight track. The graph shows the position as a function of time for this train. Note that the speed at an instant is the slope of the line at any point on the line. The graph shows that the train:
1 2 3 4
11%
22%
39%
28%
1. speeds up all the time.
2. slows down all the time.
3. speeds up part of the time and slowsdown part of the time.
4. moves at a constant velocity.
time
position
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45
![Page 34: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/34.jpg)
34
A car travels West at 20m/s. It begins to slow. Use the convention that East is +x. The acceleration of the car is considered positive since if it slowed to 19m/s in 1.0s, then
ssms
sm
t
vva o //1
1
/)20(19
Motion Diagram:
v
v(t)
a
+-
Motion Diagram Example
![Page 35: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/35.jpg)
35
Example: A car starts from rest and travels West with uniformly increasing speed. Use the convention that East is +x. Is the acceleration + or -? Is the total force acting on the car + or -? Draw a motion diagram.
Assume it goes from 0 to -10m/s in 10s.
ssms
sm
t
vva o //1
10
/)0(10
Net-force parallel to acceleration, i.e. force is – direction.
motion diagram
Net Force, Acceleration, & Motion Diagrams
![Page 36: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/36.jpg)
36
A car can accelerate at 6m/s/s. The time to go from 40mph to 60mph is:
smmi
m
s
h
h
mi/87.17
1
1609
3600
140 sm
mi
m
s
h
h
mi/81.26
1
1609
3600
160
atvv o
sssm
sm
a
vvt o 49.1
//6
/87.1781.26
Example using Acceleration
![Page 37: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/37.jpg)
37
VehicleAverage Stopping Distance at 55 mph (includes reaction time)
Passenger car 190 ft.
Tractor-trailer (loaded) with cool brakes
256 ft.
Tractor-trailer (loaded) with hot brakes
430 ft.
Tractor-trailer (empty) 249 ft.
Tractor only (bobtail) 243 ft.
![Page 38: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/38.jpg)
38
VehicleStopping Distancefrom 60 mi/hr
Accel.
feet meters ft/s2 m/s2
BMW M3 120 37 32.3 9.8
Dodge Colt GL 167 51 23.2 7.1
![Page 39: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/39.jpg)
39
![Page 40: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/40.jpg)
40
Time to Stop
BMW
st
t
atvv o
75.2
8.9270
st
t
atvv o
80.3
1.7270
Colt
![Page 41: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/41.jpg)
41
y and v graphs for tossed object in “free-fall”
![Page 42: 1 Topics Distance, Location, Speed Speed and Direction Directional quantities Acceleration Free Fall Graphs of Motion Derivatives and Integrals.](https://reader034.fdocuments.net/reader034/viewer/2022051001/56649ea05503460f94ba28ec/html5/thumbnails/42.jpg)
42
Determine how realistic 6m/s/s is for a car by computing the 0 to 60mph time:
Good time, but can be done.
sssm
sm
a
vvt o 46.4
//6
/081.26
Realistic Car?