Motion: Kinematics
Chapter 2
Scalar versus Vector Quantities
Scalar Quantities
– Magnitude (size)
– “55 mph”
Vector Quantities
– Magnitude (size)
– Direction
– “55 mph, North”
Direction Matters
Example One
1m
3m
Dx = +2 m
Example Two
1m
Dx = -3 m
4m
Speed
Two Possible Situations for Everything:
1. Object moves at a constant speed.
2. Object is being “speeded up” or “slowed
down” by some force (acceleration)
“What about an object that is just sitting there?”
Speed
Average Speed = distance (in meters)
time (in seconds)
vave =
Dx
Dt
Speed: Example One
During a 3.00 second time interval, a runner’s
position changes from x1 = 50.0 m to x2=30.5
m. What was her average speed?
(Ans: -6.50 m/s)
Speed: Example Two
A bicycle has an average speed of
5miles/hr. How far did it go in two hours?
Speed: Example Three
How long would it take a car traveling 72
km/hr to travel 60 km?
Speed: Example Four
A car is driven at an average speed of 100.0
km/hr for two hours, then driven at an
average speed of 50.0 km/hr for the next
hour. What was the average speed for the
three-hour trip? (Ans: 83.3 km/hr)
Since the speed changed, we have to address
each part of the trip separately.
1. To find the distance for the first two hours,
we use:
x=vt
x= 100km/hr X 2 hr = 200 km
2. To find the distance for the last hour, we
use the same process:
x= 50km/hr X 1 hr = 50 km
3. To find the average velocity, we will divide
the total distance by the time:
v=x/t = 250 km /3 hr = 83.3 km/hr
Velocity: Example 5
A plane flies 300.0 miles east in 2.00 hours.
The plane then turns around and flies west
for 100.0 miles in 1.00 hour.
a) Calculate total distance, total time, and
average speed. (133 mph)
b) Calculate the average velocity. (67.0 mph)
Average vs. Instantaneous Speed
Average Speed
averaged for an entire trip
Instantaneous Speed
Speed at an infinitely short interval
Constant velocity
Changing velocity
Acceleration
Acceleration – A change in velocity
– Speed (speeding up or slowing down)
Speeding up or
slowing down
– Direction (turning a bicycle)
Acceleration
• Formula
a = Dv
=
Dt
vfinal – vinitial
time
• Units = m/s2
• Acceleration can be negative (slowing down,
decelerating)
Acceleration: Example 1
What is the acceleration of a bicycle that
moves from rest to 5 m/s in 3 seconds?
Acceleration: Example 2
Superman is flying through space and
slows down from 24,000 m/s to 20,000 m/s
in 5 seconds. What is his acceleration?
What is unusual about this number?
Acceleration: Example 3
A car starts from a complete stop at a
stoplight and accelerates at 5 m/s2 for 10
seconds. What is its’ final velocity?
(Ans = 50 m/s)
Acceleration: Example 4
A car moving at 30 m/s decelerates at
–3.0 m/s2. How long will it take for it to
come to a complete stop?
(ANS: t = 10 seconds)
Kinematic Equations
(Constant acceleration only)
a = v – vo
t
x = xo + vot + ½ at2
Timeindependent
a = v 2 - v o2
2x
v = v + vo
2
Kinematics: Example 1
An airplane takes off at 27.8 m/s and can
accelerate at 2.00 m/s2. How long must the
runway be if the plane is to take off safely?
(Ans: 193 m)
Kinematics: Example 2
How long does it take a car to cross a 30.0 m
intersection, from rest, if it accelerates at
2.00 m/s2?
(Ans: 5.48 s)
Warm Up
• Determine the final velocity and time for the
Hot Wheels car as it goes down the track.
• Calculate acceleration.
• Calculate how far the car would travel on the
level in 10 seconds.
• Would it really travel this far?
Kinematics: Example 3
A pitcher throws a baseball at 44 m/s. What is
the acceleration if the ball travels through 3.5
m from the start of the pitch to the release?
(Ans: 280 m/s2)
Kinematics: Example 4
You wish to design an airbag that can protect a
driver from a head-on collision at 100 km/h.
Assume the car crumples on impact a
distance of 1.00 m.
a) What is the driver’s acceleration?
b) How fast does the airbag need to act?
(time)
Kinematics: Example 5
A driver is moving at 28 m/s. It takes him 0.50
seconds to react and slam on the brakes. At
that time, the brakes provide an acceleration
of –6.0 m/s2. Calculate the total stopping
distance.
(Ans: 79 m)
Warm Up
A horse starts from rest and can reach a speed
of 4.5 m/s in 5 seconds.
a) Calculate the average speed of the horse.
(2.25 m/s)
b) Calculate the acceleration of the horse. (0.9
m/s2)
c) Calculate the distance the horse will travel
in 35 seconds, assuming the horse keeps
accelerating. (551 m)
Aristotle versus Galileo.
In the steel cage.
Falling Objects
Aristotle (Greece, ~400 B.C.)
Heavier object fall faster than lighter ones
Galileo (1600’s, Renaissance)
All objects fall at the same rate (near the
surface of the Earth, in a vacuum)
Graph Galileo’s Numbers
Galileo found that an object is at the following heights
over time (from the Leaning Tower)
Time(s)
0
0.5
1.0
1.5
2.0
2.5
3.0
Distance(m)
56
55
51
45
36
25
12
Graph the Distance versus the Time
Distance (ft)
Distance vs. Time
Time (seconds)
Distance vs. Time
60
Distance (m)
50
40
30
20
10
0
0
0.5
1
1.5
Time (s)
2
2.5
3
The displacement
gets greater as it
falls
Acceleration due to Gravity
•
•
•
Can act at a distance, is constant near the
surface of the earth
g = 9.80 m/s2
Usually set the top of the drop at zero
yo = 0
Gravity
Why don’t people fall off the earth at the
South Pole?
Gravity
Do things accelerate forever?
• Not in an atmosphere.
• Air drag causes “terminal
velocity”
• Otherwise raindrops would
hit us with deadly speed.
• Terminal velocity
–
–
Fastest a human can drop in
air
120 miles/hr (about 60 m/s).
Gravity: Freefall
• Freefall – falling only under the acceleration
of gravity
• Object was dropped, not thrown down
y = yo + vot + ½ at2
Simplifies to:
y = ½ at2
Gravity
A rock is dropped from a cliff, calculate how far
it falls during the following times. Also,
draw a sketch of how it falls.
Time
Distance
1s
2s
3s
4s
5s
Gravity
A rock is dropped from a cliff, calculate how far
it falls during the following times. Also,
draw a sketch of how it falls.
Time
Distance
1s
4.9 m
2s
19.6 m
3s
44.1 m
4s
78.4 m
5s
123 m
0s
0m
1s
4.9 m
2s
19.6 m
3s
44.1 m
4s
78.4 m
Gravity: Example One (A)
A rock is dropped into a well and hits the water
in 2.45 seconds. How deep is the well?
(Ans: 29.4 m)
Gravity: Example One (B)
At what speed did the rock hit the bottom of the
well?
(Ans: 24.1 m/s)
Gravity: Example Two
A ball is thrown off a cliff with a initial
velocity of 3.00 m/s. Calculate its position
and speed after 1.00 s and 2.00 s.
(Ans: 7.90 m, 12.8 m/s and 25.6 m, 22.6 m/s)
A person throws a ball into the air with an initial
velocity of 15.0 m/s.
a) How high will the ball go? (Hint: use –9.80
m/s2) (Ans: 11.5 m)
b) A person throws a ball into the air with an
initial velocity of 15.0 m/s. How long is it in
the air? (Ans: 3.06 s)
c) What is the ball’s velocity when it returns to
the boy’s hand? (Ans: -15.0 m/s)
d) Does the ball have zero acceleration at its
highest point?
Gravity: Example Three (D)
What time will the ball pass a point 8.00 M
above the person’s hand?
Ans: 0.69 s and 2.37 s (you need to use the
quadratic formula)
The Lava Bomb
A volcano shots a blob of lava upward at 23.2
m/s.
a. Calculate the maximum height. (27.5 m)
b. Calculate the time to reach the maximum
height. (2.37 s)
c. Calculate the total time in the air. (4.73 s)
Arrow example
An arrow is shot into the air and reaches a
maximum height of 30.0 m.
a. Calculate the initial speed of the arrow.
(24.2 m/s)
b. Calculate the total time the arrow spent in
the air. (4.94 s)
c. Calculate the time(s) when the arrow is at
15.0 m. (0.73 s and 4.21 s)
A person stands on a cliff that is 15.0 m
from the ground. He throws a rock at 10.0
m/s straight up, but it falls down off the
cliff.
a) Calculate the maximum height. (20.1 m)
b) Calculate the time up. (1.02 s)
c) Calculate the time down (2.03 s)
d) Calculate the speed at which it will hit the
ground. (19.8 m/s)
A person stands on a building that is 25.0 m
from the ground. He throws a rock at 15.0
m/s straight up, but it falls down off the
building.
a) Calculate the maximum height. (36.5 m)
b) Calculate the time up. (1.53 s)
c) Calculate the time down (2.73 s)
d) Calculate the speed at which it will hit the
ground. (26.7 m/s)
A person standing on the ground throws a
rock at 18.0 m/s. The rock lands on top of
a roof that is 10.0 m tall.
a) Calculate the maximum height. (16.5 m)
b) Calculate the time up. (1.84 s)
c) Calculate the time down (1.15 s)
d) Calculate the speed at which it will hit the
roof. (11.3 m/s)
A person standing on the ground throws a
ball at 21.0 m/s. The ball lands on top of
a cliff that is 15.0 m tall.
a) Calculate the maximum height. (22.5 m)
b) Calculate the time up. (2.14 s)
c) Calculate the time down (1.24 s)
d) Calculate the speed at which it will hit the
cliff. (12.1 m/s)
e) Calculate the total distance the ball travels.
(30.0 m)
Graphing
• Velocity is the slope of the x vs. t graph
v = Dx = (x2-x1)
Dt
(t2-t1)
• Acceleration is the slope of the v vs. t graph
a = Dv=
Dt
(v2-v1)
(t2-t1)
(Use instantaneous rather than average velocity)
Time (s)
1
Displacement
(m)
22
2
44
3
66
4
88
5
110
6
132
7
154
8
176
9
198
10
220
Velocity (m/s)
Acceleration
(m/s2)
X
Displacement vs. Time
Displacement (m)
250
200
150
100
50
0
0
1
2
3
4
5
Time (s)
6
7
8
9
10
Velocity vs. Time
Velocity (m/s)
25
20
15
10
5
0
0
2
4
6
Time (s)
8
10
12
Acceleration vs. Time
Acceleration (m/s2)
1
0.8
0.6
0.4
0.2
0
0
2
4
6
Time (s)
8
10
12
Time (s)
Displacement (m)
0
0
0.25
0.0625
0.50
0.2500
0.75
0.5625
1.00
1.0000
1.25
1.5625
1.50
2.2500
1.75
3.0625
2.00
4.0000
2.25
5.0625
2.50
6.2500
2.75
7.5625
3.00
9.0000
Velocity (m/s)
(Use instantaneous rather than average velocity)
Acceleration
(m/s2)
Displacement (m)
Displacement vs. Time
10
9
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
Time(s)
2
2.5
3
3.5
Velocity vs. Time
7
Velocity (m/s)
6
5
4
3
2
1
0
0
0.5
1
1.5
Time (s)
2
2.5
3
3.5
Acceleration vs. Time
Acceleration (m/s2)
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
Time (s)
2
2.5
3
3.5
Graphing
Graph the following values
time(s)
displacement(m)
1
22
2
44
3
66
4
88
5
110
6
88
7
66
8
44
9
22
10
0
Displacement vs. Time
Displacement (m)
120
100
80
60
40
20
0
0
1
2
3
4
5
6
Time(s)
7
8
9
10
11
Velocity vs. Time
30
Velocity (m/s)
20
10
0
-10 0
2
4
6
-20
-30
Time(s)
8
10
12
Acceleration vs. Time
Acceleration (m/s2)
0
-10
0
2
4
6
-20
-30
-40
-50
Time(s)
8
10
12
Rules for Graphs
Slope = v
x-t graphs
Slope = a
v-t graphs
a-t graphs
Area = x
Area = v
Link to Calculus
Displacement
Slope
Area
(Derivative)
(Integrate)
Velocity
Slope
Area
(Derivative)
(Integrate)
Acceleration
Given the following distance-time graph,
calculate the velocity of the vehicle.
Given the following Velocity versus Time graph:
a. Sketch an Acceleration versus Time graph
(slope)
b. Calculate the displacement (distance)
travelled in
a. 2 seconds
c. 6 seconds
b. 4 seconds
d. 8 seconds
c. Draw a Displacement versus Time graph.
(area)
Velocity (m/s)
Velocity vs. Time
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
Time (s)
7
8
9
10
11
Acceleration vs. Time
Acceleration (m/s2)
5
4
3
2
1
0
0
2
4
6
Time (s)
8
10
12
Displacement vs. Time
Displacement (m)
250
200
150
100
50
0
-50
0
2
4
6
Time (s)
x = t2
8
10
12
Given the following Velocity versus Time graph:
a. Sketch an Acceleration versus Time graph
b. Calculate the total distance travelled at the
following times:
a. 15 s
b. 20 s
c. 25 s
d. 30 s
Velocity vs. Time
Velocity (m/s)
20
15
10
5
0
0
5
10
15
20
Time(s)
25
30
35
Given the following a-t graph, calculate the
velocity of the falling object at:
a.
b.
c.
d.
e.
1s
2s
3s
4s
5s
Kinematic Equations
v = vo + at
x = xo + vot + ½ at2
v2 = vo2 + 2ax
v = v + vo
2
2) 36 min
8 a) 10.4 m/s b) +3.5 m/s
10) 4.43 h and 881 km/h
14) 5.2 s
16) -6.3 m/s2 (0.64 g)
20) -2.4 m/s2
22) a = 4.41 m/s2, t = 2.61 s
24) 33 m/s
26) a = -390 m/s2, 40 g’s
34) 60.0 m
34) 60.0 m
36) h = 32 m, t = 5.1 s
38) 16.2 m/s, 13 m
40) vo = 4.85 m/s, t = 0.990 s
44 a) 12.8 m/s b) 0.735s, 3.35s
c) On the way up and on the way down
62) 1.3 m
52.a)
b)
c)
d)
54 a)
b)
constant v is between 0 and 17 s
t = 28 s
t = 38 s
both directions
2.5 m/s2 and 0.6 m/s2
450 m
Answers to practice quiz
1. a) 4.86 m/s2
b) 156 m c) 48.6 m/s
2. a) 5.7 m/s
b) 0.58 s c) No, g
3. a) 6.25 X 1014 m/s2 b) 7.95 X 10-9 s
c) 2.52 X 106 m/s
4. a) 1.28 m b) 0.51 s c) 5 m/s d) 11.1 m/s
5. a) 165 m/s2
b) 330 m
c) 165 m/s
6. A) 99.2 m
b) 44.1 m/s
7. A) 3.36 m/s2
b) 1.64 s
Answers to Graphing
1 a) 0
b) 5 m/s
c) No, distance not changing
d) Turns around
e)-5 m/s (opposite direction)
2 a) 2 m/s2 b) 0 m/s2
c) No, const. v
d) -4 m/s2 e) 400 m
f) 600 m
3 a) 55 s
b) 55 s onward c) 100 m/s
d) 300 m/s e) 362.5 m/s
f) 162.5 m/s
4 a) Car Y at rest b) 0.5 m/s2
c) 400 m
d) 400 m
e) at 40 s
5 a) Scott is moving in the positive direction,
Kevin, negative
b) 8 hours
c) Kevin
d) VKevin = -0.5 cm/hr VScott = 0.1 cm/hr