STRAIGHT-LINE KINEMATICS

brontidegrrrMechanics

Nov 14, 2013 (3 years and 5 months ago)

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STRAIGHT
-
LINE KINEMATICS


Any study in physics is best approached by beginning with the simplest form of a concept followed
by adding complexities, as each step is understood. The concept of motion and its analysis is
fundamental to physics; consequently,
we will begin our study with this concept.

Philosophers and scientists since the beginning of time have observed and speculated on the nature
of motion but interestingly, the concepts they developed for approximately two thousand years until
Galileo’s tim
e were completely incorrect.

Our study of motion will begin by analyzing it in its simplest form:

1)

by analyzing the motion alone without being concerned about its cause
-

kinematics.

2)

by allowing the motion to occur in a straight line only.


This is ca
lled straight
-
line kinematics. In order to study straight
-
line kinematics most effectively, we
will look at the simplest concepts first and then use them to develop the more complex concepts.

An easy way to picture straight
-
line kinematics is to use a trai
n as an analogy because its motion is
restricted to tracks and for our purposes; we can assume
-

the tracks are always in a straight line.

A) Position and displacement


In order to illustrate the concepts of position and displacement we will use a locomoti
ve moving back
and forth in front of a station as it shunts rail cars. Remember the track is straight. The train station
is our reference point for analyzing the motion of the locomotive. As we stand at the station we
notice that the locomotive is 50m t
o our left, some time later it is right in front of us, still later it is 40m
to our right, and later still it is located 10m to our left again. We could draw a diagram to represent
the various locations of the locomotive as they are measured from our ref
erence point, the station.







The x
1

indicates the first position of the locomotive, x
2

the second position, and so on. The line has
marks on it, which represent a changing position of 10 metres on each side of the reference point.
Since we need
some way of indicating left and right, “+” and “
-
“ were chosen to represent these
directions. The first position, x
1
, of the locomotive is shown as
-
50m. This means that the locomotive
was 50m to the left of the train station, our reference point. Simila
rly, x
2

= 0m, x
3

= 40m, and x
4
=
-
10m. Note that position is always measured from a reference point and always has a sign
associated with it.

If we were to analyze the motion of the locomotive, we would need to know its change in position as
it moves back
and forth in front of the train station. Any time there is a change in position of an







X
1

X
4

X
2

X
3




object we say that it has undergone displacement, that is,

Displacement = change in position

Now if we want to determine any change in position we would subtract where w
e started from where
we ended up.


d
isplacement = Δx = x
f

-

x
i

The symbol Δ (delta) is a much
-
used symbol in physics, which indicates a “change in” something
and always, subtracts the initial reading from the final reading. The displacement of the locomotive
for the first interval can now be

calculated.

Δx

= x
2

-

x
1

= 0
-

(
-
50)

= +50 m


This tells us that the locomotive moved 50m in a positive direction, which is to the right. Calculate
the displacements for the other two time intervals:

Interval two


(x
2

to x
3
)





Interval three


(x
3

to x
4
)




1)

State the displacement of each of the following changes in position.


a)
-
8 km to
-
2 km b) 2 km to
-
8 km









c) 0 km to 8 km d)
-
8 km to 20 km





e) 8 km to
-
8 km






2)

The symbol Δ has other uses. Find the chan
ge in each

a)

The temperature falls from 10

C to
-
20

C. Find ΔC




b)

The value of a stock rises from $12.00 to $17.50. Find Δs.




c)

A person on a diet goes from 60 kg to 55 kg. Express Δw





d)

A ball falls from a point 20m above the ground into a h
ole 10m deep.




Find Δx.




3. Find the displacements of each interval. Which intervals have the same displacements?

Interval

X
1

X
2


X

A

5

8


B

7

-
2


C

-
5

-
2


D






E

0

2


F

-
5

-
8


G

-
5

0



B Position
-
Time Graphs

Knowing the position of a
n object tells us something about the motion of that object but it would be
helpful to know the length of time it takes to undergo the displacement. Displacement and time can be
both illustrated at the same time on a position
-
time graph. The position
-
tim
e graph is capable of
providing us with information about displacement, average velocity, and instantaneous velocity. These
three calculations will be performed
using the sample position time graph
shown below.


1) Displacement


Displacement is det
ermined on a
Posn vs time
-15
-10
-5
0
5
10
0
5
10
15
20
time (s)
position (m)
A

B

C

D

E

F




position
-
time graph using the standard formula:

displacement = Δx =x
f

-
x
i

The final and initial positions are located on the vertical coordinate, and if you desire, a dotted line
can be drawn from the graph to the position coordinate in order

to easily identify the two
positions.

In the example graph for time interval A, x
f
= 6m and x
i
= 0m.


Determine the displacements for the time intervals:

B





C






D





E





F




Given the position time graph, determine if this represents a real life
situation?


Position vs time
0
20
40
60
0
1
2
3
4
time (hours)
position (km)
d=
x
f

-
x
i


= 6m
-

0m


= 6m




POSITION
-
TIME GRAPH FOR A ROLLING BALL

Use the graph below to answer the following questions.

Determine:

a)

Δx from 0
-
4 seconds



b)

Δx from 4
-
6 seconds



c)

Δx from 6
-
9 seconds



d)

Δx from 9
-
14 seconds



e)

Δx from 0
-
14 seconds


Posn (m)
-5
0
5
10
0
5
10
15










Describe in words, the motion of the ball.
















Position Time Graph for a car in a parking lot.

Determine

a)

Δx from 0
-
2 seconds



b)

Δx from 2
-
4 seconds



c)

Δx from 4
-
7 seconds



d)

Δx from 7
-
9 seconds



e)

Δx from 9
-
12 seconds



f)

Δx from 12
-
13 seconds



g)

Δx from 13
-
14 seconds



h)

Δx from 0
-
14 seconds




Posn (m)
-5
0
5
10
15
0
5
10
15




Position Time Graphs


Draw a pos
ition time graph for the following situations
.


1. A ball is thrown up in the air. It lands on the ground at your feet, 10 seconds later. Position is
measured in the vertical plane.


2. What if position was measured in the above example in the horizontal p
lane?


3. A man walks for 8 minutes from his home on Main St. N to the Royal Bank. He spends 2 minutes
at the ATM, and then returns home.


4. A bowling ball spends 8 minutes in the rack waiting to be thrown. It takes 4 seconds for the bal to
reach the pins
, then 20 seconds to return to the rack.


5. A girl leaves her locker by the office and starts to walk to the gym. She walks 15 m, before
realizing she forgot her gym clothes. She returns to her locker, gets the clothes, and then proceeds to
the gym.