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Acceleration and deceleration
∗
Sunil Kumar Singh
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 2.0†
Abstract
Negative vector is a relative term - not an independent concept. We completely loose the signicance
of a negative vector when we consider it in isolation. A negative vector assumes meaning only in relation
with another vector or some reference direction.
Acceleration and deceleration are terms which are generally considered to have opposite meaning. However, there is dierence between literal and scientic meanings of these terms. In literal sense, acceleration
is considered to describe an increase or positive change of speed or velocity. On the other hand, deceleration
is considered to describe a decrease or negative change of speed or velocity.
Both these descriptions are
incorrect in physics. We need to form accurate and exact meaning of these two terms. In this module, we
shall explore these terms in the context of general properties of vector and scalar quantities.
A total of six (6) attributes viz time, distance, displacement, speed, velocity and acceleration are used
to describe motion.
Three of these namely time, distance and speed are scalar quantities, whereas the
remaining three attributes namely displacement, velocity and acceleration are vectors.
Interpretations of
these two groups are dierent with respect to (ii) negative and positive sign and (i) sense of increase and
decrease.
Further these interpretations are also aected by whether we consider these terms in one or
two/three dimensional motion.
The meaning of scalar quantities is more and less clear. The scalar attributes have only magnitude and
no sense of direction. The attributes distance and speed are positive quantities. There is no possibility
of negative values for these two quantities. In general, time is also positive. However, it can be assigned
negative value to represent a time instant that occurs before the start of observation. For this reason, it is
entirely possible that we may get negative time as solution of kinematics consideration.
1 Negative vector quantities
A vector like acceleration may be directed in any direction in three dimensional space, which is dened by
the reference coordinate system. Now, why should we call vector (as shown in the gure below) represented
by line (i) positive and that by line (ii) negative? What is the qualication for a vector being positive or
negative? There is none. Hence, in pure mathematical sense, a negative vector can not be identied by itself.
∗ Version
1.14: Aug 24, 2009 4:33 am -0500
† http://creativecommons.org/licenses/by/2.0/
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Vector representation in three dimensional reference
Figure 1: Vectors in isolation have no sense of being negative.
In terms of component vectors, let us represent two accelerations
a1
and
a2
as :
a1 = −2i + 3j − 4k
a2 = −2i + −3j + 4k
Why should we call one of the above vectors as positive and other as negative acceleration? Which sign
should be considered to identify a positive or negative vector?
component form is another vector given by
− a1
Further, the negative of
a1
expressed in
:
−a1 = − (−2i + 3j − 4k)
= 2i − 3j + 4k
So, what is the actual position? The concept of negative vector is essentially a relative concept. If we
represent a vector
A
A is just another vector,
A and has equal magnitude as that of A.
as shown in the gure below, then negative to this vector which is directed in the opposite direction to that of vector
A
In addition, if we denote -
as B,
then A
a negative vector when we consider it in isolation.
is -
B.
We, thus, completely loose the signicance of
We can call the same vector either A
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A.
or -
conclude, therefore, that a negative vector assumes meaning only in relation with another vector.
We
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Vector representation in three dimensional reference
Figure 2: Vectors in isolation have no sense of being negative.
In one dimensional motion, however, it is possible to assign distinct negative values. In this case, there
are only two directions; one of which is in the direction of reference axis (positive) and the other is in
the opposite direction (negative). The signicance of negative vector in one dimensional motion is limited
to relative orientation with respect to reference direction.
In the nutshell, sign of vector quantity in one
dimensional motion represents the directional property of vector.
It has only this meaning.
We can not
attach any other meaning for negating a vector quantity.
i
It is important to note that the sequence in -5 " is misleading in the sense that a vector quantity can not
i
have negative magnitude. The negative sign, as a matter of fact, is meant for unit vector . The correct
i
i
reading sequence would be 5 x - , meaning that it has a magnitude of 5 and is directed in - direction
i.e. opposite to reference direction. Also, since we are free to choose our reference, the previously assigned
i
i
-5 can always be changed to 5 and vice-versa.
We summarize the discussion so far as :
•
•
There is no independent meaning of a negative vector attribute.
In general, a negative vector attribute is dened with respect to another vector attribute having equal
magnitude, but opposite direction.
•
In the case of one dimensional motion, the sign represents direction with respect to reference direction.
2 Increase and decrease of vectors quantities
The vector consist of both magnitude and direction. There can be innite directions of a vector. On the
other hand, increase and decrease are bi-directional and opposite concepts.
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Can we attach meaning to a
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phrase increase in direction or decrease in direction? There is no sense in saying that direction of the
moving particle has increased or decreased. In the nutshell, we can associate the concepts of increase and
decrease with quantities which are scalar not quantities which are vector. Clearly, we can attach the sense
of increase or decrease with the magnitudes of velocity or acceleration, but not with velocity and acceleration.
For this reason, we may recall that velocity is dened as the time rate of change not increase or
decrease in displacement. Similarly, acceleration is dened as time rate of change of velocity not increase
or decrease in velocity. It is so because the term change conveys the meaning of change in direction as
well that of change as increase or decrease in the magnitude of a vector.
However, we see that phrases like increase or decrease in velocity or increase or decrease in acceleration
are used frequently. We should be aware that these references are correct only in very specic context of
motion. If motion is unidirectional, then the vector quantities associated with motion is treated as either
positive or negative scalar according as it is measured in the reference direction or opposite to it. Even in
this situation, we can not associate concepts of increase and decrease to vector quantities. For example, how
would we interpret two particles moving in negative x-direction with negative accelerations
2
−20m/s
−10m/s2
and
respectively ? Which of the two accelerations is greater acceleration ? Algebraically, -10 is
greater than -20. But, we know that second particle is moving with higher rate of change in velocity. The
second particle is accelerating at a higher rate than rst particle. Negative sign only indicates that particle
is moving in a direction opposite to a reference direction.
Clearly, the phrases like increase or decrease in velocity or increase or decrease in acceleration are
correct only when motion is "unidirectional" and in "positive" reference direction. Only in this restricted
context, we can say that acceleration and velocity are increasing or decreasing. In order to be consistent
with algebraic meaning, however, we may prefer to associate relative measure (increase or decrease) with
magnitude of the quantity and not with the vector quantity itself.
3 Deceleration
Acceleration is dened strictly as the time rate of change of velocity vector. Deceleration, on the other hand,
is acceleration that causes reduction in "speed". Deceleration is not opposite of acceleration. It is certainly
not negative time rate of change of velocity. It is a very restricted term as explained below.
We have seen that speed of a particle in motion decreases when component of acceleration is opposite
to the direction of velocity.
In this situation, we can say that particle is being decelerated.
Even in this
situation, we can not say that deceleration is opposite to acceleration. Here, only a component of acceleration
is opposite to velocity not the entire acceleration. However, if acceleration itself (not a component of it) is
opposite to velocity, then deceleration is indeed opposite to acceleration.
If we consider motion in one dimension, then the deceleration occurs when signs of velocity and acceleration are opposite. A negative velocity and a positive acceleration mean deceleration; a positive velocity and
a negative acceleration mean deceleration; a positive velocity and a positive acceleration mean acceleration;
a negative velocity and a negative acceleration mean acceleration.
Take the case of projectile motion of a ball. We study this motion as two equivalent linear motions; one
along x-direction and another along y-direction.
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Parabolic motion
Figure 3: The projected ball undergoes deceleration in y-direction.
For the upward ight, velocity is positive and acceleration is negative. As such the projectile is decelerated
and the speed of the ball in + y direction decreases (deceleration). For downward ight from the maximum
height, velocity and acceleration both are negative. As such the projectile is accelerated and the speed of
the ball in - y direction increases (acceleration).
In the nutshell, we summarize the discussion as :
•
•
Deceleration results in decrease in speed i.e magnitude of velocity.
In one dimensional motion, the deceleration is dened as the acceleration which is opposite to the
velocity.
Example 1: Acceleration and deceleration
Problem : The velocity of a particle along a
straight line is plotted with respect to time as
shown in the gure. Find acceleration of the particle between OA and CD. What is acceleration at
t = 0.5 second and 1.5 second. What is the nature of accelerations in dierent segments of motion?
Also investigate acceleration at A.
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Velocity time plot
Figure 4
Solution :
Average acceleration between O and A is given by the slope of straight line OA :
aOA =
v2 −v1
t
=
0.1−0
1
= 0.1m/s2
Average acceleration between C and D is given by the slope of straight line CD :
aCD =
v2 −v1
t
=
0−(−0.1)
1
= 0.1m/s2
Accelerations at t = 0.5 second and 1.5 second are obtained by determining slopes of the curve
at these time instants. In the example, the slopes at these times are equal to the slope of the lines
OA and AB.
Instantaneous acceleration at t = 0.5 s :
a0.5 = aOA = 0.1m/s2
Instantaneous acceleration at t = 1.5 s :
a1.5 = aAB =
v2 −v1
t
=
0−0.1
1
= −0.1m/s2
We check the direction of velocity and acceleration in dierent segments of the motion in order
to determine deceleration.
To enable comparision, we determine directions with respect to the
assumed positive direction of velocity. In OA segment, both acceleration and velocity are positive
(hence particle is accelerated).
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In AB segment, acceleration is negative, but velocity is positive
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(hence particle is decelerated). In BC segment, both acceleration and velocity are negative (hence
particle is accelerated).
In CD segment, acceleration is positive but velocity is negative (hence
particle is decelerated).
Alternatively, the speed increases in segment OA and BC (hence acceleration); decreases in
segments AB and CD (hence deceleration).
We note that it is not possible to draw an unique tangent at point A. We may draw innite
numbers of tangent at this point. In other words, limit of average acceleration can not be evaluated
at A. Acceleration at A, therefore, is indeterminate.
4 Graphical interpretation of negative vector quantities describing motion
4.1 Position vector
We may recall that position vector is drawn from the origin of reference to the position occupied by the body
on a scale taken for drawing coordinate axes. This implies that the position vector is rooted to the origin of
reference system and the position of the particle. Thus, we nd that position vector is tied at both ends of
its graphical representation.
r
r
Also if position vector ` ' denotes a particular position (A), then - denotes another position (A'), which
is lying on the opposite side of the reference point (origin).
Position vector
Figure 5
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4.2 Velocity vector
The velocity vector, on the other hand, is drawn on a scale from a particular position of the object with its
v
tail and takes the direction of the tangent to the position curve at that point. Also, if velocity vector ` '
v
denotes the velocity of a particle at a particular position, then - denotes another velocity vector, which
is reversed in direction with respect to the velocity vector,
v.
Velocity vector
Figure 6
In either case (positive or negative), the velocity vectors originate from the position of the particle and
are drawn along the tangent to the motion curve at that point. It must be noted that velocity vector,
v, is
not rooted to the origin of the coordinate system like position vector.
4.3 Acceleration vector
Acceleration vector is drawn from the position of the object with its tail. It is independent of the origin
(unlike position vector) and the direction of the tangent to the curve (unlike velocity vector). Its direction
is along the direction of the force or alternatively along the direction of the vector representing change in
velocities.
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Acceleration vector
Figure 7
a
a
a.
Further, if acceleration vector ` ' denotes the acceleration of a particle at a particular position, then - denotes another acceleration vector, which is reversed in direction with respect to the acceleration vector,
Summarizing the discussion held so far :
1: Position vector is rooted to a pair of points i.e. the origin of the coordinate system and the position
of the particle.
2:
Velocity vector originates at the position of the particle and acts along the tangent to the curve,
showing path of the motion.
3: Acceleration vector originates at the position of the particle and acts along the direction of force or
equivalently along the direction of the change in velocity.
4: In all cases, negative vector is another vector of the same magnitude but reversed in direction with
respect to another vector. The negative vector essentially indicates a change in direction and not the change
in magnitude and hence, there may not be any sense of relative measurement (smaller or bigger) as in the
case of scalar quantities associated with negative quantities. For example, -4
than +4
◦
◦
C is a smaller temperature
C. Such is not the case with vector quantities. A 4 Newton force is as big as -4 Newton. Negative
sign simply indicates the direction.
We must also emphasize here that we can shift these vectors laterally without changing direction and
magnitude for vector operations like vector addition and multiplication. This independence is characteristic
of vector operation and is not inuenced by the fact that they are actually tied to certain positions in the
coordinate system or not. Once vector operation is completed, then we can shift the resulting vector to the
appropriate positions like the position of the particle (for velocity and acceleration vectors) or the origin of
the coordinate system (for position vector).
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