**Kinematics - Linear Motion**

1.1

**Linear motion**and**Non-Linear Motion**:
·

**Linear motion**is motion in a**straight line –**For examples: A passenger is carried by an escalator; or an athlete running a 100 m race; whereas
·

**Non-linear motion**is motion**not in a straight line -**A top spinning; or the earth orbiting the Sun.
1.2

**Kinematics**and**Dynamics:**
·

**Kinematics**is the study of the**motion**of an object**without**considering the**force**acting on it. Therefore, the equations of motion do not have force F as a variable in them.
·

**Dynamics**is the study of**motion**and the**forces**acting on the object.
1.3
The

**physical quantities**involved in**linear motion:**
·
Distance and Displacement

·
Speed and Velocity

·
Acceleration and Deceleration

·
Time

**:**

__The Physical Quantities Explained__
Suppose an object at point O moves
east 100 metres (m) to point A in 14 seconds (s); then it immediately moves
west from point A to point B 40 m away in 8 s:

·

**:**__Distance__
o The
total

**distance**moved by the object = (100 + 40) m = 140 m;
o

**Distance**is the**total length**of the path travelled by an object in motion from 1 position to another position;
o

**Distance**is a**scalar quantity**– it does not take into account the direction of motion.
o In
linear motion: Distance = Magnitude of Displacement (since direction of motion
remains constant)

·

**:**__Displacement__
o The
total

**displacement**of the object from point O = (100 – 40) m = 60 m east of point O;
o

**Displacement**is a measure of**how far**and**the direction**in which an object has been**displaced****from**a reference point (e.g.**original position**) due to the motion
o

**Displacement**measures the straight-line distance and the direction between the initial position and the final position of an object due to the motion.
o

**Displacement**is a**vector quantity**because it has both the**magnitude**and the**direction**.
·

__Speed__:
o The
object’s

**Average Speed**from O to B = 140 m / 22 s = 6.36 m s^-1:
§
Its

**Av. Speed**from O to A = 100 m / 14 s = 7.14 m/s
§
Its

**Av. Speed**from A to B = 40 m / 8 s = 5 m/s
o Thus,

**Average Speed**, v, = Total Distance Travelled, s (m) / Time Taken, t (s).
o

**Constant Speed**: If an object**moves equal distances**in**equal time**intervals, then it is moving with**constant speed**–**otherwise**, it is moving with**non-uniform speed**.
o

**Speed**is a measure of**how fast**an**object moves –**the**rate of distance travelled**in the motion.
o

**Speed**is a**scalar quantity –**it**measures only**the**magnitude of distance**moved**over time**with**no regard to direction**of motion.
o In
linear motion: Speed = Magnitude of Velocity (since the direction of motion
remains constant)

·

__Velocity__:
o Its

**average velocity**from O to B = (100 – 40) m / 22 s or 2.73 m/s due east of O:
§
Its

**Av. Velocity**from O to A = 100 m / 14 s = 7.14 m/s due east of O;
§
Its

**Av. Velocity**from A to B = - 40 m / 8 s = - 5 m/s (west of A)
o Thus,

**Average Velocity**, v = Total Displacement,**s**(m) / Time Taken, t (s).
o Constant
Velocity

**v**means equal displacements in equal time intervals.
o

**Constant Velocity**: If an object**moves with equal displacement**in**equal time**intervals, then it is moving with**constant velocity**–**otherwise**, it is moving with**non-uniform velocity**.
o

**Velocity**is a measure of how fast an object is displaced (the rate of displacement of the object) from a reference position (its initial position).
o

**Velocity,**unlike speed,**is a****vector quantity**because it measures both the**magnitude**and**direction**of displacement due to motion.
o

**A change in velocity**means either a change in speed or a change in direction (or both) of motion.
·

__Speed and Velocity in Linear Motion__:
o

**In a linear motion**, the direction of motion remains unchanged: The speed of linear motion equals to the magnitude of velocity since the distance travelled per unit time in a linear motion is the same as the displacement per unit time along the straight line. Thus, in linear motion:**v**= v = Total distance travelled, s (m) / Time taken, t (s)

o

**In linear motion**, a**change in velocity**can only**means**a**change in the speed**with**no change****in direction**of motion and this change of speed can only be**due to acceleration**(increase in speed over time) or**deceleration**(decrease in speed over time).
·

__Acceleration and Deceleration__:
o

**Acceleration**is defined as the**rate of change of velocity**(or rate of change of speed, in linear motion):
Acceleration, a

**=****Change in Velocity / Time Taken**
= (Final Velocity, v –
Initial Velocity, u) / Time, t

a = (v – u) / t …

**1st Equation of Motion**
o

**Acceleration is positive**if**initial velocity, u, increases**with**time, t,**to final velocity**, v:**that is,**if v > u, a is +ve**.
o Conversely,

**acceleration is negative**if**velocity decreases**to final velocity**, v,**from initial velocity, u: that is,**if v < u, a is –ve**, in which case, it is known as**deceleration**or**retardation.**
o

**Constant**(or uniform)**acceleration:**
·
If an object moves with

**equal change**in**velocity**(or speed in linear motion) in**equal time**intervals, then it is moving with**constant acceleration**–**otherwise**, it is moving with**non-uniform acceleration**.
·
An object moves with constant

**acceleration**, a, for**time**, t, will change the**initial velocity**, u, to**final velocity**, v, as follows:
v = u + at …

**2nd Equation of Motion;**Therefore,
t = (v – u) / a …

**3rd. Equation of Motion**
o

**Zero acceleration:**If**velocity**is**constant**or uniform (or, speed is constant or uniform in the case of linear motion), the**acceleration is zero**since change in velocity (or, speed) is zero.
o

**Acceleration**or deceleration is a**vector quantity**with SI unit metre per second per second, m s^-2.
o

**Equations of Motion**are derived on the basis of constant or zero acceleration – please see later.

·

__Time (Scalar Quantity in second (s)):__
·
Time interval is an important quantity in the
study of motion – it is measured by a stopwatch or by the use of
“Ticker-Timer”.

·
There are 2 types of stopwatches depending of
the accuracy needed:

__Accuracy Needed__

__Type of Stopwatch Used__

0.1 s ~ 0.2 s Analogue
(mechanically-operated)

0.01 s Digital (electronically-operated)

·
Apart from stopwatch, another device that we use
to measure time interval of linear motion is the “ticker-timer” – please see
the ensuing.

·
We know that an object that moves with constant

**acceleration**, a, for**time**, t, will change its**initial velocity**, u, to**final velocity**, v:**v = u + at**…

**2nd Equation of Motion**(as in foregoing);

Therefore, time,

**t = (v – u) / a**…**3rd Equation of Motion**
·

__Ticker-Timer__**Ticker-timer**is a device that can be used to determine a number of quantities relating to linear motion of an object (including time interval), namely:

a)
Time Interval of the Motion

b)
Displacement of the Object

c)
Velocity of the Object

d)
Acceleration of the Object

e)
Type of Motion of the Object

·
The above quantities can be determined because
ticker-timer has a metal strip with a pin that vibrates up and down at 50 Hz
(which is the frequency of the 12 V or 6 V ac power supply). Each time the pin
moves down at the interval of 1/50 seconds (or 0.02 s), it makes a dot on the
pre-carbonated ticker tape which passes beneath it as the tape is pulled by the
moving object to which the tape is attached. Thus,

a)
The

**time interval**which elapses**between successive dots**is 1/50 second or**0.02 s**(T = 1/f) – therefore, the time interval of the motion between any 2 points / dots can be determined by:**Multiplying**the**number of dots**after the 1^{st}point / dot until the other point / dot**by****1/50 s**; Therefore:
1 dot-space (1-tick space) = 1 x
0.02 s = 0.02 s

2 dot-space (2-tick of time) = 2 x
0.02 s = 0.04 s

5 dot-space (5-tick of time) = 5 x
0.02 s = 0.1 s

10 dot-space (10-tick space) = 10
x 0.02 s = 0.2 s

b)

**Displacement**of the object**between any 2 points**can also be determined by measuring the**distance between the 2 points**on the ticker tape
c)

**Velocity**:
·
The

**velocity**of the objects**between 2 points**/ dots is the displacement**between the 2 points**/ dots**over**the number of**tick-time**over the 2 points / dots;
·
The

**average velocity**between any 2 points is the**total****displacement**between the 2 points of the tape**over**the**total time**intervals elapsed (i.e. the number of tick-time) between the 2 points.
·

**Constant**(or, Uniform)**Velocity:**When the ticker tape shows**equal displacement**over**equal time intervals,**the object is moving linearly with**constant**or uniform**velocity**.
d)

**Acceleration**or**Deceleration**:
·
The

**acceleration**(or deceleration) of the objects between 2 successive intervals of motion on the ticker-tape is the change in velocities between the 2 intervals over the time between the mid-points of the 2 intervals. The velocities can be determined as in the foregoing.
·
The

**average****acceleration**(or deceleration) of a moving object between any 2 intervals of motion (note: intervals not points) can be determined by comparing the**corresponding****velocities**for the 2 intervals over the time between the mid-points of the 2 intervals -**increasing velocity**means**acceleration**and decreasing velocity means**deceleration.**
·

**Constant**(or uniform)**Acceleration:**When the ticker tape shows**equal increase**(or decrease)**in velocity**over**equal time interval**, the object is moving linearly with**constant acceleration**(or constant deceleration, which ever is appliacable)
e)
From the foregoing, it is therefore clear: That the

**type of motion**of an object – whether it is moving linearly at**constant velocity**(zero acceleration), at**irregular acceleration**or deceleration or**at constant acceleration**– can be seen and determined from the pattern of the dots on the ticker tape.
·

__Equations of Motion__

We have already learnt from the
foregoing 3 basic

**equations of linear motion**where the object moves with**constant**(or uniform)**acceleration**, a, with**initial velocity**, u,**final velocity**, v, for**time**, t:
1)

**a = (v – u) / t**…**1**^{st}**Equation of Motion**
2)

**v = u + at**…**2**and^{nd}Equation of Motion;
3)

**t = (v – u) / a**…**3**.^{rd}**Equation of Motion 3**
The

**4**is about displacement or distance travelled, s, of the object under the same state of motion:^{th}Equation of Motion
4)
s = Average Velocity x Time Taken

**s = ½ (u + v)t**…

**4**

^{th}Equation of Motion

The

**5**are obtained from the 4^{th}and 6^{th}Equations of Motion^{th}Equation [s = ½ (u + v)t] by substituting the 2^{nd}Equation (v = u + at) and the 3^{rd}equation [t = (v - u)/a] respectively into the 4^{th}Equation. Thus,
5) s = ½ (u + v)t…from 4

^{th}Equation
s = ½ [u + (u + at)]t …(substitute
2

^{nd}equation into the 4^{th})
s = ½ [2u + at]t

**s = ut + ½ at^2**…

**5**

^{th}Equation of Motion
(or, s = 1/2 (g) (t^2), free fall under gravity:

**2005 P1 Q4. pg 3**)
6)
s = ½ (u + v)t…from 4

^{th}Equation
s = ½ (u + v)(v – u)/a …(substitute 3

^{nd}equation into the 4^{th})
s = ½ (v^2 – u^2)/a …(Form 3 algebraic
expansion)

2as = v^2 –
u^2

**v^2 = u^2 + 2as**…

**6**

^{th}Equation of Motion
Science is best learnt by
understanding rather than by memorizing formulae (rote learnig). I believe, by
understanding alone, you should be able to easily derive the

**first 4 equations of motion**. In solving a kinematics problem, just ask yourself this: What is the variable that the question wants me to find the value?
· If acceleration,

**a**: then, use**1**;^{st}equation
· If final velocity,

**v**: use**2**;^{nd}equation
· If time,

**t**: use**3**;^{rd}equation
· If displacement,

**s**: use**4**.^{th}equation
At times, you need to find the
value of another variable before you can solve your problem using the above simplified
method – ample examples in my handouts.