x is the magnitude of a position

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Variables used in kinematics
x is the magnitude of a position v is the
magnitude of the velocity, sometimes speed a
is the magnitude of acceleration t
is time ? represents a change, an
interval. ?x is displacement, and can be also
called d sometimes ?v is a change in speed
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Distance or displacement?
This is a vector. It is a number AND a direction.
There are special rules to operate with vectors.
It is not the same as a simple number.
This is a scalar. It is a number, a distance,
doesnt matter the direction of the trajectory.
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Whats the difference between speed and velocity?
Speed is a scalar. Scalars have only a magnitude
(just a number). For example constant speed of a
car at 60 km/h.
Velocity is a vector. Vectors have a magnitude
AND a direction. For example velocity
of an airplane traveling 200 km/h Northward.
60 km/h
Speed can be the magnitude of the velocity.
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A car goes around a curve at constant speed. Is
the cars velocity changing?

• Yes
• No

At position A, the car has the velocity indicated
by the arrow (vector) v1. At position B, the car
has the velocity indicated by the arrow (vector)
v2, with the same magnitude (speed) but a
different direction.
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Our frame of reference (it is a necessary
convention)
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Quick Quizzes
Answer True or False
F
1) A car must always have an acceleration in the
same direction as its velocity.
T
2) Its possible for a slowing car to have a
positive acceleration.
3) An object with constant nonzero acceleration
can never stop and remain at rest.
T
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This is a table showing the positions of car 1
and car 2 for a motion in 5 seconds.
Which one moves further away in the same
time? Which one has the greatest average speed?
time xcar1 xcar2 __(s)
(m) (m)
0 0 0 1 10 20 2 20 40 3 30 60 4 40 80 5 50
100
Lets plot this on the blackboard.
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EQUATION OF A STRAIGHT LINE
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What does a cars speedometer measure?

• Average speed
• Instantaneous speed
• Average velocity
• Instantaneous velocity

A speedometer measures instantaneous speed.
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Which quantity is the highway patrol more
interested in?

• Average speed
• Instantaneous speed

The speed limit indicates the maximum legal
instantaneous speed. To estimate the time a trip
may take, you want to use average speed.
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Passing lane
Watch the red car
Watch the blue car
POSITION versus TIME
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In a graph POSITION versus TIME, the SLOPE
indicates VELOCITY. A steeper slope indicates
greater velocity.
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Passing lane
Watch the red car
Watch the blue car
VELOCITY versus TIME
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In a graph VELOCITY versus TIME, constant
velocity is a straight line.
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In general, Average ? Instantaneous
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Is the instantaneous velocity at point A greater
or less than that at point B?

1. Greater than
2. Less than
3. The same as
4. Unable to tell from this graph

The instantaneous velocities can be compared by
looking at their slopes. The steeper slope
indicates the greater instantaneous velocity, so
the velocity at A is greater.
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In the graph shown, during which time interval is
the acceleration greatest?

• Between 0 s and 2 s.
• Between 2 s and 4 s.
• Between 4 s and 8 s.
• The acceleration does
• not change.

The acceleration is greatest between 2 s and 4 s.
The velocity is changing fastest, and the graph
has the greatest slope, during this interval.
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At which point is the magnitude of the
acceleration the greatest?

• Point A
• Point B
• Point C
• The acceleration does not
• change.

The magnitude of the acceleration is greatest
when the velocity is changing the fastest (has
the greatest slope). This occurs at point A.
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2.5 CONSTANT ACCELERATION
Important equations!
Many applications of mechanics involve objects
moving with constant acceleration. Constant a
means that average a instantaneous a For
convenience, lets adopt vf v and vi v0 ,
ti 0 , tf t so
or
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Because velocity is increasing or decreasing
UNIFORMLY with time, we can express
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final
average
So now I will substitute inside ,
and then again inside the resulting
equation.
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Remember that or
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1
3
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Graph v vs t
Graph x vs t
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From a graph slope or area?
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What should you do if you have a velocity versus
time graph and you want.
acceleration ?
displacement ?
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During which time interval is the distance
traveled by the car the greatest?

• Between 0 s and 2 s.
• Between 2 s and 4 s.
• Between 4 s and 6 s.
• It is the same for all time
• intervals.

The distance traveled is greatest when the area
under the velocity curve is greatest. This
occurs between 2 s and 4 s, when the velocity is
constant and a maximum.
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Important our equations can only be used in
regions of CONSTANT ACCELERATION.
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Constant velocity in the positive direction
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Constant velocity in the negative direction
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CONSTANT POSITIVE ACCELERATION
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CONSTANT NEGATIVE ACCELERATION
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UNIFORM MOTION (velocity is constant)
UNIFORM ACCELERATION (acceleration is constant)
t
v
Velocity
v(t)
v0
Slope 0
t
0
a
Acceleration
a(t) 0
t
0
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Quick Quiz
Parts (a), (b), and (c) of the figure below
represent three graphs of the velocities of
different objects moving in straight-line paths
as functions of time. The possible accelerations
of each object as functions of time are shown in
parts (d), (e), and (f). Match each velocity vs.
time graph with the acceleration vs. time graph
that best describes the motion.

1. a and e, b and f, c and d
2. a and d, b and f, c and e
3. a and e, b and d, c and f

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(a)
Match this plot
5
4
5
Velocity (m/s)
3
4
2
Distance (m)
3
1
2
0
0
1
2
4
10
6
8
Time (s)
0
0
2
4
10
(b)
6
8
Time (s)
5
(d)
4
5
Acceleration (m/s2)
3
4
Velocity (m/s)
2
3
1
2
0
1
0
2
4
10
6
8
Time (s)
0
0
2
4
10
6
8
Time (s)
(c)
(e)
3
5
2
4
Velocity (m/s)
Acceleration (m/s2)
3
1
0
2
-1
1
-2
0
0
0
2
4
10
2
4
10
6
8
6
8
Time (s)
Time (s)
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(a)
60
Match this plot
50
40
2
Distance (m)
30
1
Acceleration (m/s2)
20
0
10
-1
0
0
-2
2
4
10
6
8
Time (s)
8
0
2
4
10
(b)
6
8
Time (s)
6
(d)
4
5
Velocity (m/s)
2
4
Velocity (m/s)
0
3
-2
2
1
0
2
4
10
6
8
Time (s)
0
0
2
4
10
6
8
Time (s)
(c)
(e)
20
.
30
15
25
Distance (m)
10
Distance (m)
20
5
15
0
10
-5
5
0
2
4
10
6
8
0
2
4
10
6
8
Time (s)
Time (s)
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Acceleration of gravity does not depend on the
mass
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Gravitational acceleration g does NOT depend on
the weight of the object.
Apollo 15 Moon walk in 1971. Commander David
Scott
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Equations governing falling objects and objects
thrown upward
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What is the balls acceleration at the top of its
path (at t2 s)?

1. zero.
2. 9.8 m/s
3. -9.8 m/s
4. 9.8 m/s2
5. -9.8 m/s2

Gravity does not turn off at the top! The
balls velocity is still changing, as it changes
from going up to going down. For a moment the
velocity is zero, but the gravitational
acceleration is a constant throughout the path.
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Always
t 2 s y 20 m v 0
a -g
Maximum height ymax 20 m
Lets suppose now that the initial velocity of
the ball upward is 35 m/s (instead of the 20
m/s as in the picture). What is the maximum
height that the ball will reach? Assume that g
10 m/s2
t 1 s y 15 m v 10 m/s
t 3 s y 15 m v 10 m/s
Equations we can use
t 0 y 0 v 20 m/s
t 4 s y 0 m v 20 m/s
Answer ymax 61 m
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Quick Quizzes
A tennis player on serve tosses a ball straight
up. While the ball is in free fall, its
acceleration
1. increases. 2. decreases. 3. increases and then
decreases. 4. decreases and then increases. 5.
remains constant.
A tennis player on serve tosses a ball straight
up. As the tennis ball travels through the air,
its speed
1. increases. 2. decreases. 3. increases and then
decreases. 4. decreases and then increases. 5.
remains constant.
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