Explain clearly, with examples, the distinction between:
magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;
(a) Magnitude of displacement is the shortest distance between the initial and final positions of a particle in a given time interval. It is a straight line joining the two positions. Total path length is the actual distance travelled by the particle along the path during the same interval.
For example, consider a particle moving from point A to point B (distance 10 m) and then returning to point A. The magnitude of displacement is zero (since initial and final positions are the same), but the total path length is 20 m.
Thus, total path length is always greater than or equal to the magnitude of displacement. The equality holds when the particle moves in a straight line in one direction without turning back.
magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true? [For simplicity, consider one-dimensional motion only].
(b) Magnitude of average velocity is defined as the magnitude of displacement divided by the time interval: |v_avg| = |displacement| / Δt. Average speed is defined as the total path length divided by the time interval: Average speed = Total path length / Δt.
Since total path length is always greater than or equal to the magnitude of displacement, it follows that average speed is always greater than or equal to the magnitude of average velocity.
For example, if a car travels from Delhi to Agra (200 km) in 4 hours and returns to Delhi in another 4 hours, the total path length is 400 km, but displacement is zero. Average speed = 400/8 = 50 km/h, but average velocity = 0/8 = 0 km/h.
The equality holds when motion is in a straight line without change in direction.
Explanation
This question tests understanding of the fundamental difference between scalar quantities (distance, speed) and vector quantities (displacement, velocity). The key insight is that path length accounts for the entire journey, while displacement only considers the net change in position. Students must recognize that equality occurs only in rectilinear motion without reversal.