Chapters 1 and 2 (Motion I and II) Summary
The three fundamental quantities are length (unit : m), time (unit : s) and mass (unit : kg).
To measure a time interval using a stop watch, an error is introduced due to the time lapse between seeing the
event and starting (or stopping) the watch. This error ( 誤差 ) is called the reaction time (反應時間). Reaction
time varies from person to person but is, on the average, about 0.2 s. Stop-watch is not suitable for measuring
time intervals of about one second as the reaction time introduces too great an error.
Displacement ( 位移 ) is defined as the change in position of a point in a specified direction.
Distance is a scalar quantity ( 無向量 ) which has magnitude ( 大小 ) but no direction. Other examples of
scalar quantities are mass, area, speed, energy and temperature.
Displacement is a vector quantity ( 向量 ) which has both magnitude and direction. Displacement is the
distance travelled in a specified direction. Other examples of vector quantities are velocity, acceleration and
force.
A vector is represented graphically by a straight line and an arrow. Using a suitable
scale, the length of the line represents the magnitude of the vector and the arrow
gives its direction. Two vectors are said to be equal if they are of the same length
------->
B
------->
q
p
and direction. The two vectors shown are equal. They are denoted by AB and CD
-->
D
-->
( or p and q for short ).
------->
------->
------->
------->
A
C
Magnitude of AB = AB ; magnitude of CD = CD 
The resultant (合量) or vector sum of several vectors is a vector which would produce the same effect as the
original ones.
Addition of vectors
1. If the vectors are collinear, the resultant can be found by simple algebraic addition.
2. If the vectors are not collinear, the resultant can be found by tip-to-tail method and parallelogram method as
follows :
Tip-to-tail method
The "tip" of the first vector is joined to the "tail" of the second vector (and so on if there are more than two
vectors). The resultant is represented by a line from the "tail" of the first to the "tip" of the second vector
(the last if there are more than two).
Parallelogram method
The diagonal of the parallelogram represents the resultant.
Note the following:
1. For scalars, 3 + 4 = 7. 7 is called algebraic sum. But for vectors, 3 + 4 = unknown unless the angle between
the two vectors are given. If 3 and 4 are in the same direction, the vector sum is 7 , having its maximum
value 7. If 3 and 4 are in opposite direction, the vector sum is 1 or -1 (depending on the direction you take
as positive), having its minimum value. If 3 and 4 are at right angle, the vector sum is 5 (by Pythagorus
theorem). As a summary, for vectors,
7 (maximum value when 3 + 4 are in the same direction
3+4 =
1 (minimum value when 3 + 4 are in opposite direction
5 (when 3 + 4 are at right angle
1.1
2. A - B is defined as A + -B where -B is defined as the vector which has the same magnitude as B but exactly
opposite direction as B.
3. Apart from displacement, the following are also vectors: velocity (not speed, speed is scalar), forces, weight
etc. So their addition must be performed using tip-to-tail method or parallelogram method.
Resolution of vectors
Two vectors can be summed up and replaced by a single resultant vector which is called the vector sum of these
vectors. The resultant vector produces the same effect as the two vectors. Conversely, it is possible to replace a
single vector by two vectors. These vectors are called components (分量) of the original vector. The splitting of
a vector into its components is called resolution (分解) of vector.
In physics problems, we always resolve a vector into two components which are perpendicular to each other.
The reason of this is that the two components are then independent of each other.
As an example, the single force F is resolved into two components, one in x-direction and the other in ydirection.
From geometry,
x-component of F = Fx = F cos 
y-component of F = Fy = F sin 
Note the following:
1. The components are smaller in magnitude than the original vector in this case.
But in general, components can be larger than resultant.
2. The resultant of the components must, of course, be the original vector.
We can use sine or cosine to find the vector sum of two vectors, say 5 N and 3 N as follows:
Sum of horizontal components = 5 cos 35o- 3 cos 30o = 1.50
o
5N
o
Sum of vertical components = 5 sin 35 + 3 sin30 = 4.37
3N
Resultant =
1.50  4.37  4.62
2
2
s 
Velocity is the rate of change of displacement.  v 

t 

30o
35o
Speed is the rate of change of distance traveled.
v 
Acceleration is the rate of change of velocity.  a 

t 

if an object continues to travel at the same speed, it is accelerating if its direction
changes. For example, an object whirling around a circle at a constant speed is
accelerating since its direction of travel changes all the time, although its speed does
not change. That is to say, the direction of (tangential) velocity (represented by
direction of arrows) always changes from time to time, and so the velocity changes
from time to time, i.e. accelerating. Only when both the direction and magnitude of
velocity do not change, then the body is undergoing uniform motion in a straight line.
1.2
P
Q
average accelerati on 
change of velocity 
v 
a 

time for the change 
t 
When  t becomes very small, the average acceleration approaches the acceleration at a particular time interval.
The unit of acceleration is m s-2.
For uniform acceleration a along a straight line, it can be shown that
a =
v
vu
=
t
t
where a = acceleration, v = final velocity, u = initial velocity, t = time taken.
Graphical interpretation
1. Displacement-time graph
(a) Velocity at time t ' = slope of the curve at t '
In an s-t graph, slope of tangent = velocity
(b) The graph represents motion with increasing velocity (because the slope of tangent is increasing).
2. Velocity-time graph
(a) The acceleration at any time t ' = slope of tangent at t '
(b) The area under the graph between time t1 and t2 = displacement between time t1 and t2.
In a v-t graph,
slope of tangent = acceleration,
area under graph = displacement
1.3
The three equations of uniformly accelerating motion (等加速運動方程式) are as follows:
v  u  at - - - - - - - - - - without s
1
s  ut  at 2 - - - - - - - without v
2
2
2
v  u  2as - - - - - - - without t
Note the following :

The above equations are applicable only when the acceleration is constant. That is to say, the motion must
be uniformly accelerating motion.

u , v and a are vectors. When two or more vectors of different directions are considered (i.e. they appear in
the same equation), we must take one direction to be positive and the opposite direction to be negative.
When these three equations are used, always remember the following rule: the direction of initial velocity
is usually taken as positive. (初速永遠是正的)

Strictly speaking, s is displacement. But in cases where the linear motion is in one direction only, i.e. the
object does not turn backwards, s can be distance travelled, i.e. then the magnitude of displacement is
equal to the distance travelled. In this case, velocity can be expressed as rate of change of distance
travelled.

How is the direction of acceleration a related to velocity v and ‘change of velocity v ’? The answer is
that a has no definite relation with v but a and  v must be in the same direction. For example,
when an object is thrown upward, the acceleration is downward (because force of gravity is downward),
velocity is upward and change of velocity (defined as final velocity – initial velocity) is also downward
(because the final velocity vector is of smaller magnitude than the initial velocity vector).

If a body is travelling at constant velocity, it must be travelling at constant speed and in a straight line.
But, conversely, if a body is travelling at constant speed, it may not necessarily travel in a straight line.
Uniform circular motion is an example.
Acceleration of freely falling bodies
All freely falling bodies fall with the same acceleration due to gravity 10 m s -2 provided that air resistance is
negligible.
1. For a body which is projected upwards, the time it takes to reach the highest point is the same for it to
return from the highest point to the original starting point.
2. For an object which is projected upwards, at the highest point, the speed of the object is zero. We say that
the object is momentarily at rest ( 瞬時靜止 ). But note that its acceleration at the highest point is still g =
10 m s-2 downwards (The explanation is that the force which causes this acceleration is the weight, which
is unchanged). This fact requires some time to understand. If you do not understand now, just remember the
fact. It is easier for you to understand this fact after studying the next chapter.
A motion sensor works like a radar. It emits ultrasound signals and detects its echoes from a moving object.
From the signal-echo time lag, the computer quickly works out the information about the motion, and generates
s-t, v-t graph and a-t graphs.
In using the motion sensor, it must be noticed that the position reading is always positive. Hence, the direction
away from the sensor is taken as positive, i.e. the position reading (displacement) is more positive as the object
moves away from the motion sensor. Therefore,
For an s-t graph,
positive velocity (positive slope) corresponds to motion away from the motion sensor.
Negative velocity (negative slope) corresponds to motion towards the sensor.
1.4