SHE1215 1 CHAPTER 2 KINEMATICS IN ONE DIMENSION Study Objectives: At the end of the class, students should be able to a) define distance, displacement, speed, velocity and acceleration b) state the difference between vector and scalar quantities c) solve problems using equations of motion with constant acceleration d) sketch and interpret graphs of displacement-time, velocity-time and acceleration- time for motion of a body e) solve problems using equations of motion with constant acceleration to analyze free fall. Mechanics: the study of how objects move and the forces that causes motion. Dynamics: the branch of physics that studies force and the causes of various types of motion. Kinematics: the branch of physics that describes motion of objects without considering the effects that produce the motion. The motion of objects can be explained by using words, diagrams, numbers, graphs, and equations. The goal of any study of kinematics is to develop sophisticated mental models which serve to describe (and ultimately, explain) the motion of real- world objects. 2.1 DISPLACEMENT Motion is related to change of position. Distance and displacement are two quantities which may seem to mean the same thing, yet they have distinctly different meanings and definitions. Distance – The actual path length between two points/length measured along the path line. - Distance is a scalar quantity (magnitude with no direction) Displacement
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SHE1215
1
CHAPTER 2
KINEMATICS IN ONE DIMENSION
Study Objectives:
At the end of the class, students should be able to
a) define distance, displacement, speed, velocity and acceleration
b) state the difference between vector and scalar quantities
c) solve problems using equations of motion with constant acceleration
d) sketch and interpret graphs of displacement-time, velocity-time and acceleration-
time for motion of a body
e) solve problems using equations of motion with constant acceleration to analyze free
fall.
Mechanics: the study of how objects move and the forces that causes motion.
Dynamics: the branch of physics that studies force and the causes of various types of motion.
Kinematics: the branch of physics that describes motion of objects without considering the
effects that produce the motion. The motion of objects can be explained by using words,
diagrams, numbers, graphs, and equations. The goal of any study of kinematics is to develop
sophisticated mental models which serve to describe (and ultimately, explain) the motion of real-
world objects.
2.1 DISPLACEMENT
Motion is related to change of position.
Distance and displacement are two quantities which may seem to mean the same thing, yet they
have distinctly different meanings and definitions.
Distance
– The actual path length between two points/length measured along the path line.
- Distance is a scalar quantity (magnitude with no direction)
Displacement
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– The displacement is a vector that points from an object’s initial position to its final
position and has a magnitude that equals the shortest distance between the two positions.
– The direct straight line pointing from the initial point to the final point / the change in position
of the object.
- Displacement is a vector quantity (has both magnitude & direction)
- The displacement: 0xxx
- + sign: direction to the right/east
- - sign: direction to the left/west
- SI Unit : meter (m)
Fig 2.1 The arrow represents the displacement x2 x1 = x .
Figure 2.1 The displacement x is a vector that points from the initial position x0 to the final
position x.
EXAMPLE 2.1:
Distance = 3.0 m + 4.0 m = 7.0 m
Displacement = 5.0 m
C
5.0 m
A 3.0 m
B
4.0 m
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EXAMPLE 2.2
A boy runs 30 m east, 40 m north, 50 m west. What is his net displacement?
SOLUTION :
mmmmd 45)3050()40( 22
01 2740
3050tan
m
mm west of north
AVERAGE SPEED – indicates how fast an object is moving.
The average speed of an object is the total distance traveled by the object per unit time.
Average speed = timeElapsed
traveldistance total
Speed is a scalar quantity.
SI unit : m/s
AVERAGE VELOCITY - indicates how fast an object is moving and the direction of its
motion.
Defined as displacement of the object divided by the time interval during which the
displacement occurred.
Average velocity = takentime
ntdisplaceme of change
t
x
tt
xx
0
0
Average velocity is a vector quantity and its direction follows that of the change in displacement.
SI unit : m/s
50 m
30 m
40 m
d
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INSTANTANEOUSLY VELOCITY
- is the velocity (magnitude & direction) of an object at a particular instant of time ( t is closed
to zero)
t
x
t
0lim
SI unit : m / s
Instantaneous Speed - speed at any given instant in time.
Average Speed - average of all instantaneous speeds; found simply by a distance/time ratio.
EXAMPLE 2.3 :
A plot of position versus time is in Fig. 1 for an object in linear motion.
gure 1
(a) What are the average velocities for the segments AB,BC,CD,DE,EF,FG, and BG?
(b) State whether the motion is uniform or non uniform in each case.
(c) What is the instantaneous velocity at point D?
SOLUTION:
(a) if
if
tt
xx
t
x
000.1
0.10.1
s
mmAB
smss
mmBC /0.3
0.10.3
0.10.7
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smss
mmCD /3.1
0.35.4
0.70.9
smss
mmDE /3.1
5.40.6
0.90.7
smss
mmEF /7.1
0.60.9
0.70.2
00.90.11
0.20.2
ss
mmFG
smss
mmBG /10.0
0.10.11
0.10.2
b) The motion of BC, CD, and DE are not uniform since they are not straight lines.
c) The object changes its direction of motion at point D. So it has to stop momentarily and
0 .
ACCELERATION
When the velocity of an object changes in magnitude, or in direction, or in both magnitude and
direction, the object is said to accelerate.
Acceleration is the rate of change of velocity.
Average acceleration the change in velocity divided by the time interval to make the
change.
timeelapsed
yin velocit changeonaccelerati average
0
0
ttta
Acceleration is a vector quantity and its direction follows the direction of the change in velocity.
The direction of the acceleration vector depends on two factors:
whether the object is speeding up or slowing down
whether the object is moving in the positive (+) or negative (–) direction
a = 0 (zero acc ) ---velocity is a constant
a = + (positive acc) --- acceleration (speed increase)
a = - (negative acc) --- deceleration (speed decrease)
Instantaneous acceleration is the acceleration of an object at a particular instant of time.
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SI unit : m/s2
The direction of the instantaneous acceleration follows the direction of the instantaneous velocity
change.
2.2 Equations Of Kinematics For Constant Acceleration (Uniform acceleration)
A body can have many ways of motion: in one dimensional path, curve, circular, parabolic and
many others. This chapter will only discuss the motion on one straight line with a constant
acceleration.
Kinematics variables: 1) x = displacement
2) a = acceleration
3) = final velocity at time t
4) 0 = initial velocity at time t0=0 s
5) t = time elapsed since t0=0 s
Kinematics Equations:
The equations of kinematics apply when an object moves with a constant acceleration along a
straight line. These equations relate the displacement x-x0, the acceleration a, the final velocity
, the initial velocity 0 , and the elapsed time t-t0.
Assuming that x0=0 m at t0=0 s, the equations of kinematics are:
1) at 0
2) 2
0
3) tx )(2
10
4) 2
02
1attx
5) ax22
0
2
The process to determine unknown information about an object's motion involves the use of a
problem-solving strategy which includes the following steps:
1. Construct an informative diagram of the physical situation.
2. Identify and list the given information in variable form.
3. Identify and list the unknown information in variable form.
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4. Identify and list the equation which will be used to determine the unknown information
from the known variables.
5. Substitute known values into the equation and use appropriate algebraic steps to solve for
the unknown.
6. Check your answer to ensure that it is reasonable and mathematically correct.
EXAMPLE 2.4
A car accelerates from 13m/s to 25m/s in 6.0s.What was it acceleration? How far did it travel in
this time? Assume constant acceleration.
Solution:
By definition, the acceleration is 20
25m s 13m s2.0m s
6.0 s
v va
t
.
The distance of travel can be found from Eq. 2-11b.
22 21 1
0 0 2 213m s 6.0 s 2.0m s 6.0 s 114 mx x v t at
EXAMPLE 2.5 A car slows down from 23m/s to rest in a distance of 85m. What was it acceleration, assumed constant?
Solutions:
22 2
2 2 20
0 0
0
0 23m s2 3.1m s
2 2 85 m
v vv v a x x a
x x
.
EXAMPLE 2.6 A world class sprinter can burst out of the blocks to essentially top speed(of about 11.5m/s) in the first
15.0m of the race. What is the average acceleration of this sprinter, and how long does it take her to reach
that speed?
Solutions:
The sprinter starts from rest.
22 2
2 2 2 20
0 0
0
11.5m s 02 4.408m s 4.41m s
2 2 15.0 m
v vv v a x x a
x x
.
The elapsed time is found by solving Eq. 2-11a for time.
0
0 2
11.5m s 0 2.61 s
4.408m s
v vv v at t
a
2.3 Motion Diagrams / Graphical Analysis
- Graphical techniques are often helpful in understanding motion and its related quantities
especially velocity and acceleration.
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bmxy : ta 0
Graph : position versus time (x-t) --- velocity (slope)
Velocity versus time (v-t) --- acceleration (slope)
(A) Displacement graphs
Case 1: Positive velocity, constant velocity:
To begin, consider a car moving with a constant, rightward (+) velocity of 10 m/s.
Note that a motion with constant, positive velocity results in a line of constant and positive slope
when plotted as a position-time graph.
Figure (a) : Positive Velocity Constant Velocity
Conclusions:
1) Instantaneous velocity is the same at any time.
2) The gradient is positive which means positive velocity. Horizontal line, velocity is zero.
Case 2: Positive velocity, changing velocity (acceleration)
Now consider a car moving with a changing, rightward (+) velocity – that is, a car that is moving
rightward and speeding up or accelerating.
If the position-time data for such a car were graphed, the resulting graph would look like the
graph as shown above. Note that a motion with changing, positive velocity results in a line of
changing and positive slope when plotted as a position-time graph.