Animated Solution for Physics - Kinematics: A particle moves in a circle of radius $R$. In half revolution its displacement is $\dots \dots$ and distance covered is $\dots \dots$ .

Imagine standing at the edge of a vast, perfectly circular arena. You are about to witness one of the most fundamental concepts in physics: the distinction between the journey and the destination.

Our problem introduces a particle that embarks on a journey along a circular path of radius $R$. It doesn't complete a full lap; instead, it travels exactly half a revolution. To truly understand the physics of this motion, we must view it through two entirely different lenses: the lens of Distance and the lens of Displacement.

The Journey Matters

Distance is the story of the journey. It accounts for every twist, every turn, and every inch of ground covered along the path. It is a scalar quantity, meaning it possesses only magnitude. Think of it as the reading on your car's odometer—it only ever goes up, regardless of which direction you drive.

When the particle embarks on its journey from its starting point, let's call it point $A$, it traces the elegant curve of the circle. A full circle represents a complete revolution. The total path length of this full journey is the circumference of the circle, given by the timeless geometric formula:

However, our particle is a traveler of moderation. It halts its journey exactly halfway, at a point we will call $C$, passing through the top of the circle at point $B$. This half-revolution means it has traced exactly half of the full circumference. Mathematically, we express this as:

This elegant result, $\pi R$, represents the actual physical ground covered by the particle as it moved along the semicircular arc.

The Crow's Flight

Now, let us shift our perspective from the journey to the destination. Displacement is the cold, hard truth of the result. It is the crow's flight. It asks only two questions: Where did you start? And where did you end up?

Displacement is a vector quantity, armed with both magnitude and direction. It cares nothing for the beautiful arc the particle just traced. It demands the shortest, most direct route from the genesis, point $A$, to the terminus, point $C$.

If we draw a straight line from $A$ to $C$, we notice a profound geometric truth. Because the particle completed exactly half a revolution, points $A$ and $C$ lie on exactly opposite ends of the circle. The straight line connecting them must, by the laws of geometry, pass directly through the center of the circle, point $O$.

A line segment that connects two points on a circle and passes through the center is known as the diameter. The diameter is composed of two radii: the segment from $A$ to $O$, and the segment from $O$ to $C$.

Substituting the radius $R$ for both segments, we arrive at the magnitude of the displacement:

The direction of this displacement vector is simply from point $A$ directly towards point $C$.

This simple problem serves as a profound masterclass in kinematics. It beautifully illustrates how two different physical quantities can describe the exact same event in completely different ways.

The distance ($\pi R$) tells us about the effort expended, the actual path taken through space. The displacement ($2R$) tells us about the net result, the absolute change in position.

Because the shortest path between two points is always a straight line, the magnitude of displacement will never exceed the distance traveled. In our circular arena, the curved path ($\pi R\approx 3.14R$) is indeed longer than the direct shortcut ($2R$).

By mastering this distinction, you lay the unbreakable foundation for all future studies in mechanics, from the parabolic arcs of projectiles to the elliptical orbits of planets.