Lesson on d-t and v-t graphs for linear motion

1. Position-Time Graphs &
Velocity-Time Graphs
Also look at uniform motion and constant
velocity, page 18.
Reference pages: 18, 19-20, 27-29, 32

2. Uniform Motion
 Before we start to look atBefore we start to look at dd–t–t graphs, andgraphs, and vv-t-t
graphs, we are going to define two more motiongraphs, we are going to define two more motion
terms.terms.
 Constant velocityConstant velocity is constant inis constant in bothboth magnitudemagnitude
andand directiondirection. In the velocity vector,. In the velocity vector, vv = 3.5 m/s= 3.5 m/s
[E], the[E], the 3.5 m/s3.5 m/s is called theis called the magnitudemagnitude andand [E][E]
is the direction.is the direction.
 Uniform motionUniform motion means that an object moves withmeans that an object moves with
constant velocity.constant velocity.

3. Position-Time GraphsPosition-Time Graphs
 Consider the graph. TheConsider the graph. The
magnitude of the velocitymagnitude of the velocity
is represent by the _____is represent by the _____
 What type of motion doesWhat type of motion does
the object have in regionsthe object have in regions
A, B, and C ?A, B, and C ?
 Uniform MotionUniform Motion
 Can we say the objectCan we say the object
moves with uniformmoves with uniform
motion from 0 to 14s?motion from 0 to 14s?
 NoNo
P o s it io n T im e G r a p h
2 4 6 8 1 0 1 2 1 4
Position(m)
2468101214
T im e ( s )
A
B C

4.  If we wanted to find theIf we wanted to find the
average velocity from 0 toaverage velocity from 0 to
12s, how would we do this?12s, how would we do this?
 From the graph, we canFrom the graph, we can
see that we would needsee that we would need
to get the slope of the lineto get the slope of the line
drawn from the start ofdrawn from the start of
interval A to the end ofinterval A to the end of
interval Binterval B
P o s it io n T im e G r a p h
2 4 6 8 1 0 1 2 1 4
Position(m)
2468101214
T im e ( s )
A
B C
A
&
B
 For vFor v avgavg for intervals A, B, and C we wouldfor intervals A, B, and C we would
draw a line from the origin to the end of C anddraw a line from the origin to the end of C and
find the slopefind the slope

5.  Consider a car driving at aConsider a car driving at a
constantconstant 200 km/h around200 km/h around
an oval track. Can we sayan oval track. Can we say
that the car is moving withthat the car is moving with
uniform motion?uniform motion?
 No, since the direction isNo, since the direction is
constantly changingconstantly changing
Finding Instantaneous VelocityFinding Instantaneous Velocity
 From the graph we can see theFrom the graph we can see the
slope is constantly changing. Soslope is constantly changing. So
how would we find thehow would we find the
instantaneous velocity at pointinstantaneous velocity at point AA
oror B?B?
P o s it io n T im e
G r a p h
Position(m)
T im e ( s )
A
B
T o p V ie w
( L o o k in g D o w n )
D O M

6.  To find theTo find the
instantaneous velocityinstantaneous velocity
at pointat point A,A, we wouldwe would
draw a tangent line atdraw a tangent line at
AA and find the slopeand find the slope
of the tangent.of the tangent.
 If we wanted to findIf we wanted to find
vvavgavg between pointsbetween points
AA andand BB we would:we would:
 Find the slope ofFind the slope of
the line throughthe line through
both points.both points.
Linejoining
A
andB
P o s i t i o n T i m e
G r a p h
Position(m)
T i m e ( s )
A
B
T
an
g
en
t
L
in
e
P o s i t i o n T i m e
G r a p h
Position(m)
T i m e ( s )
A
B

7. Drawing a v-t Graph From a d-t GraphDrawing a v-t Graph From a d-t Graph
 To find the velocityTo find the velocity
from the first graph,from the first graph,
we would have to:we would have to:

Find the slope ofFind the slope of
each part of theeach part of the
graph. Since wegraph. Since we
do not have ado not have a
scale, we willscale, we will
approximate theapproximate the
magnitudes ofmagnitudes of
velocity.velocity.
P o s it io n T im e G r a p h
Position(m)
T im e ( s )
V e l o c i t y T i m e G r a p h
Velocity(m/s)
T im e ( s )

8. Practice Problems
 Page 30 #11Page 30 #11
 Page 53 # 33, 34, 40, 58Page 53 # 33, 34, 40, 58
 WB page 15WB page 15 (New page # 17) QuestionQuestion
#12#12
{{Clarify reference point and starting point
on a d–t graph. }}
 WB page 16WB page 16 (New page #18) # 7# 7