Posts
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 1st
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 …1st Equation
of Motion
2)
v = u +
at …2nd Equation of
Motion; and
3)
t = (v –
u) / a …3rd. Equation of Motion 3
The 4th Equation of Motion is
about displacement or distance travelled, s, of the object under the same state
of motion:
4)
s = Average Velocity x Time Taken
s
= ½ (u + v)t … 4th Equation
of Motion
The 5th and 6th Equations of Motion are obtained
from the 4th Equation [s = ½ (u + v)t] by substituting the 2nd
Equation (v = u + at) and the 3rd equation [t = (v - u)/a]
respectively into the 4th Equation. Thus,
5) s = ½ (u + v)t…from 4th Equation
s = ½ [u + (u + at)]t …(substitute
2nd equation into the 4th)
s = ½ [2u + at]t
s = ut + ½ at^2 …5th
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 4th Equation
s = ½ (u + v)(v – u)/a …(substitute 3nd
equation into the 4th)
s = ½ (v^2 – u^2)/a …(Form 3 algebraic
expansion)
2as = v^2 –
u^2
v^2 =
u^2 + 2as … 6th 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 1st equation;
· If final velocity, v: use 2nd
equation;
· If time, t: use 3rd equation;
· If displacement, s: use 4th 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.
