2.2 Acceleration and deceleration  (Page 4/4)

Graphical interpretation of negative vector quantities describing motion

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 find that position vector is tied at both ends of its graphical representation.

Also if position vector ‘ r ’ denotes a particular position (A), then “- r ” denotes another position (A’), which is lying on the opposite side of the reference point (origin).

Position vector

Velocity vector

The velocity vector, on the other hand, is drawn on a scale from a particular position of the object with its 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 “- v ” denotes another velocity vector, which is reversed in direction with respect to the velocity vector, v .

Velocity vector

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.

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.

Acceleration vector

Further, if acceleration vector ‘ a ’ denotes the acceleration of a particle at a particular position, then “- a ” denotes another acceleration vector, which is reversed in direction with respect to the acceleration vector, a .

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° C is a smaller temperature than +4° 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 influenced 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).