Question

Can the displacement vector for a particle moving in two dimensions be longer than the length of the path traveled by the particle over the same time interval? Can it be less? Discuss.

Short answer

The displacement vector cannot be longer than the distance but can be less.

Step-by-step solution

Step 1. Definition of distance and displacement 

Distance is a scalar quantity and is defined as the total path covered by an object.

Displacement is a vector quantity and can be defined as the shortest straight line distance between the beginning and the ending points.

Distance never decreases with time and can never be zero or negative. However, displacement can take up any value; zero, positive, or negative.

Step 2. When a particle travels along a straight path

If a particle moves along a straight path between any two points, the length of the displacement vector will be the same as the distance traveled by the particle.

Step 3. When a particle travels along a non-linear path

If the path taken by a particle is not straight or consists of different line segments, the length of the displacement vector will be less than the distance traveled by the particle.

Thus, the displacement vector can never be longer than the path length, although it can be equal or shorter.