Meaning of vector attributes

The scalar scheme assigns a reference direction. The attributes are given positive or negative values depending on whether they are in the direction of reference or opposite to it. This underlines the meaning of the sign of the vector attributes. The “sign” represents direction. According to this scheme, it is perfectly valid to represent a vector attribute as scalar variable like “a” and assign negative sign as in :

a = - 5 m / s 2 a = - 5 m / s 2

The important aspect of the sign in the above expression on the right of the equation is that it is not indicative of direction on its own. The sign indicates direction in conjunction with a given reference direction. We can always reorient the reference direction such that the sign of the attribute is reversed. For example, a velocity in one reference, v = 5 m/s becomes v = -5 m/s in the other reference with opposite direction.

Figure 1: The sign of vector attribute depends on the orientation of reference direction.

Representation of velocity

This dependence on a specific reference direction has a significant consequence : we can not represent a physical direction (like north or south unless reference is aligned in these directions), based on the sign of an attribute. For this reason, we need to specify physical direction in words, if required.

We must emphasize that negative vector attributes like negative displacement, negative velocity and negative acceleration simply means that the corresponding vector attribute is directed in opposite direction to the reference.

We conclude the discussion so far as :

Meaning of “increase” and “decrease” for vector attributes

Vector attributes are represented as signed scalars. Hence, it is possible to think about the change of these attributes in terms of increase or decrease (recall that such notion of increment is not valid for vectors). For this reason, it is valid to talk about change of vector attributes in terms of “increase” or “decrease” for one dimensional case. Thus, it is perfectly fine to say that velocity is increasing or decreasing - though the meaning may differ a bit from the conventional meaning, which conveys an associated increase or decrease in the magnitude of the quantity.

A positive increment (for example displacement increment, Δx > 0) means that either a positive vector attribute (say displacement of 5 m/s) has become more positive (say displacement of 10 m/s) or a negative vector attribute (say displacement of -10 m/s) has become less negative (say displacement of -5 m/s) with the passage of time. This aspect is illustrated in displacement – time plot for the motion of a ball along x-reference as shown in the figure.

Figure 2

Motion of a ball in one dimension
Subfigure 2.1 Subfigure 2.2

On the other hand, a negative increment (for example displacement increment, Δx < 0) means that either a positive vector attribute (say displacement of 10 m/s) has become less positive (say displacement of 5 m/s) or a negative vector attribute (say displacement of -5 m/s) has become more negative (say displacement of -10 m/s) with the passage of time. This aspect is illustrated in displacement – time plot for the motion of a ball along x-reference as shown in the figure.

Figure 3

Motion of a ball in one dimension
Subfigure 3.1 Subfigure 3.2

It must be clearly understood that the meaning of increment of attribute is not same as the meaning of increment of corresponding magnitude. For example, according to the meaning as stated, a negative increment of attribute may mean that a negative displacement of -5 m/s has become more negative (say velocity of -10 m/s) as shown in the plot above. We notice here that magnitude of displacement, as a matter of fact, has increased from 5 m to 10 m. Thus, displacement has decreased, but its magnitude has actually increased.

To illustrate the point explicitly for a real time motion under gravity where motion reverses direction, let us consider the motion of a vertically projected ball with initial velocity 30 m/s.

Figure 4: A ball projected upward involves reversal of direction.

Motion under gravity

Considering constant acceleration due to gravity, g = 10 m / s 2 m / s 2 , the velocities at successive seconds are :

---------------------------------------------
Time      Velocity     Magnitude of Velocity                         
(s)        (m/s)              (m/s)             
---------------------------------------------
0           30                 30
1           20                 20
2           10                 10
3            0                  0
4          -10                 10
5          -20                 20
6          -30                 30
---------------------------------------------

The corresponding velocity – time plot is :

Figure 5: Note that slope is negative and constant for the motion irrespective of the direction of motion.

Motion under gravity : Velocity time plot

The velocity increment , Δv, from t = 0 s to t = 2 s is negative :

Δ v = v 2 - v 1 = 10 - 30 = - 20 m / s Δ v = v 2 - v 1 = 10 - 30 = - 20 m / s

The magnitude of velocity in this time interval decreases by 20 m/s. Thus, we see that a negative increment, apart from indicating a decrease in velocity, also conveys that magnitude of velocity has decreased. This is in accordance with the conventional meaning. We also note that the negative increment, Δv, conveys that a positive velocity has become less positive from 30 m/s to 10 m/s.

Now, consider increment from time t = 4 s to t = 6 s. The velocity increment, Δv, for this case also is negative :

Δ a = v 2 - v 1 = - 30 - ( - 10 ) = - 20 m / s Δ a = v 2 - v 1 = - 30 - ( - 10 ) = - 20 m / s

In this case, however, the magnitude of velocity in the time interval increases by 20 m/s. Thus, we see that a negative increment does not indicate that the magnitude of velocity has decreased. This is not in accordance with the conventional meaning. Here, we note that the negative increment, Δv, conveys that a negative velocity has become more negative from -10m/s to -30 m/s.

Further, we see that velocity increment, Δv, for the whole journey remains negative, indicating that motion involves decrease of velocity irrespective of whether magnitude of velocity decreases for one part of motion (upward motion) and increases for another part of motion (downward motion). This fact of negative velocity decrement is collaborated by the slope of the plot, which is negative for the entire motion and is equal to negative constant acceleration due to gravity.

We conclude the discussion so far as :

Meaning of vector attribute with respect to other attributes

Velocity is the time rate of change in displacement and acceleration is time rate change in velocity. Thus, the three vector attributes, namely displacement, velocity and acceleration are connected to each other. Here, we need to emphasize that we investigate the meaning of vector attribute and its increment with respect to other vector attribute. Further, we substantiate our study with the same example of the projection of a ball in the vertical direction with initial velocity of 30 m/s.

Velocity

The velocity is time rate of change in displacement. The question that we seek to answer here is that whether directions of two quantities (velocity and displacement) are same or not? In term of defining differential equation,

v = ⅆ x ⅆ t v = ⅆ x ⅆ t

A plain reading of this equation suggests that a positive displacement increment should yield positive velocity and a negative displacement increment should yield a negative velocity. As a matter of fact, this is the case.

We note from the tabulated data that increment in displacement, Δx, and velocity are both positive in the first half of the motion; and negative in the second half motion. The two vector attributes are completely synchronized with respect to direction. This is expected from the defining equation.

----------------------------------------------------
Time      Displacement    Displacement     Velocity       
                          Increment/s                         
(s)          (m)            (m)             (m/s)           
----------------------------------------------------
0             0             30               30         
1            25             25               20                
2            40             15               10                
3            45              5                0                
4            40             -5              -10                
5            25            -15              -20                
6             0            -25              -30                
----------------------------------------------------

The directional correspondence between displacement and velocity is also collaborated by displacement – time plot, where slope of the line is equal to velocity. Note that slope (i.e. velocity) of the curve is positive as the displacement increases with time for upward motion and is negative as displacement decreases for the downward motion.

Figure 6: Note that slope is negative and constant for the motion irrespective of the direction of motion.

Motion under gravity : Velocity time plot

However, the point of distinction to be made here is that direction of displacement and velocity need not be same. We see in the second half of the motion, when ball descends from the maximum height, that velocity is negative (directed against the reference direction of the coordinate system), whereas displacement is positive. This result is not contrary to the defining equation of velocity as it relates velocity with the increment in displacement (dx or Δx) – not the displacement (x).

It is also imperative from the plot and data that a positive velocity means that either a positive displacement has become more positive or a negative displacement has become less negative. Similarly, a negative velocity means that either a positive displacement has become less positive or a negative displacement has become more negative.

Further, we can also draw conclusion about the change in the magnitude of displacement from the relative directions of two attributes as :

1. If displacement and velocity are in the same direction (have same signs; either both are positive or both are negative), then magnitude of displacement increases.
2. If displacement and velocity are in the opposite direction (have different signs), then magnitude of displacement decreases.

We conclude the discussion so far as :

• A positive increment in displacement means positive velocity and a negative increment in displacement means negative velocity.
• The sign of displacement and velocity during motion are unrelated and need not be same.
• A positive velocity means that either a positive displacement has become more positive or a negative displacement has become less negative.
• A negative velocity means that either a positive displacement has become less positive or a negative displacement has become more negative.
• If displacement and velocity are in the same direction (have same signs; either both are positive or both are negative), then magnitude of displacement increases.
• If displacement and velocity are in the opposite direction (have different signs), then magnitude of displacement decreases.

Acceleration

Acceleration is related to velocity in the form of differential equation,

a = ⅆ v ⅆ t a = ⅆ v ⅆ t

A plain reading of this equation suggests that a positive velocity increment should yield positive acceleration and a negative velocity increment should yield a negative acceleration. As a matter of fact, this is the case.

The directional correspondence between displacement and velocity is also collaborated by velocity – time plot for the motion under gravity, where slope of the plot is equal to acceleration. The slope is negative corresponding to negative constant acceleration due to gravity.

Figure 7: Note that slope is negative and constant for the motion irrespective of the direction of motion.

Motion under gravity : Velocity time plot

Just as in the case of velocity, the point of distinction to be made is that direction of velocity and acceleration need not be same. We see in the first half of the motion as the ball ascends from the ground, velocity is positive but acceleration is negative. In the second half, velocity and acceleration both are negative. This result is not contrary to the defining equation of acceleration as it relates velocity with the increment in velocity (dv or Δv) – not the velocity (v).

It is also imperative from the plot that a positive acceleration means that either a positive velocity has become more positive or a negative velocity has become less negative. Similarly, a negative acceleration means that either a positive velocity has become less positive or a negative velocity has become more negative.

Further, we can also draw conclusion about the change in the magnitude of velocity from the relative directions of velocity and acceleration as enumerated here :

1. If velocity and acceleration are in the same direction (have same signs; either both are positive or both are negative), then magnitude of velocity increases.
2. If velocity and acceleration are in the opposite direction (have different signs), then magnitude of velocity decreases.

The above conclusions point to an interesting aspect of deceleration. We have seen that deceleration is special type of acceleration, which results in reduction in the magnitude of speed. The second paradigm of relative directions between velocity and acceleration ensures this. It is evident that velocity and acceleration must be in the opposite direction (have different signs)for deceleration.

We conclude the discussion so far as :

• A positive increment in velocity means positive acceleration and a negative increment in velocity means negative acceleration.
• The sign of velocity and acceleration during motion are unrelated and need not be same.
• A positive acceleration means that either a positive velocity has become more positive or a negative velocity has become less negative.
• A negative acceleration means that either a positive velocity has become less positive or a negative velocity has become more negative.
• If velocity and acceleration are in the same direction (have same signs; either both are positive or both are negative), then magnitude of velcocity increases.
• If velocity and acceleration are in the opposite direction (have different signs), then magnitude of velcocity decreases.