Distance vs. Displacement

Motion is the movement of a body. If a particle moves in a two dimensional coordinate system, that is in a plane, both of its coordinates change and the motion is two dimensional motion. If the particle movies in space, that is in a three dimensional coordinate system, the motion of the particle is three dimensional.

In everyday life you may think the distance and displacement are the same things but they are totally different in physics. Distance is a scalar quantity and displacement is a vector quantity. Being scalar distance does not have any particular direction associated with it but displacement has direction.

Figure 1 The displacement is the straight line between the initial and final position which has a direction towards the final position; it does not go into the details of the path of motion. But distance is the total length of travel whether that is a curve or a straight line.

In Figure 1 a particle moves from the initial position $s_1$ to the final position $s_2$ along the curved path. In this case the distance represents the whole path of motion but the displacement only depends on the initial and final positions.

Therefore, the displacement is the straight line joining the two positions which has direction towards the final position from the initial position (as the particle reached the final position).

You may ask, can distance and displacement be the same? The similarity of distance and displacement is that the distance can also be equal to the magnitude of displacement in a particular situation, that is if the motion is along a straight line without a turn. But distance and displacement are never the same. The distance is always a positive quantity but displacement can be positive or negative. The SI unit of distance and displacement is meter (m).

The above figure illustrates an example of distance and displacement. Another example of distance and displacement is, you moved along a circular path and returned to the initial position. The displacement is zero, as the initial and final positions are the same but the distance is the whole path of motion.

In other words the displacement does not depend on the details of motion of the particle. If you consider this information in a coordinate system and if the initial position vector of point $p_1$ is $\vec r_1$ and the final position vector of point $p_2$ is $\vec r_2$, the equation (aka formula) of displacement is

\[\Delta \vec r = \vec r_2 - \vec r_1\]

A position vector is simply a vector representing a particular position in a coordinate system. If the motion is along a straight line, the distance is the magnitude of displacement but beware that the magnitude of displacement is not equal to distance if the motion is not along a straight line as illustrated by the Figure 1 above.

Mechanics
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