Areal velocity is the rate at which a body sweeps out area as it moves along its path. It's a concept used in physics, particularly when dealing with central forces.

Here's why areal velocity is constant for a body moving under a central force:

1. Conservation of Angular Momentum

* A central force is a force that always points towards a fixed point (the center of force). This means the force has no component perpendicular to the radius vector connecting the body to the center.

* In the absence of external torques, angular momentum is conserved.

* For a central force, the torque about the center of force is zero because the force is radial. Therefore, the angular momentum of the body is conserved.

2. Relating Angular Momentum and Areal Velocity

* Angular momentum (L) of a body of mass (m) moving with a velocity (v) at a distance (r) from the center of force is given by: L = mvr sin θ, where θ is the angle between the velocity and the radius vector.

* The area swept out by the body in a small time interval (dt) is approximately half the area of the parallelogram formed by the radius vector and the displacement vector (v dt).

* This area is given by: dA = (1/2) r (v dt sin θ)

* Therefore, the areal velocity (dA/dt) is: dA/dt = (1/2) rv sin θ

3. Connecting the Dots

* Comparing the expressions for angular momentum and areal velocity, we see that:

* L = 2m (dA/dt)

* Since angular momentum (L) is conserved, the areal velocity (dA/dt) is also constant.

In simpler terms:

* Imagine a planet orbiting a star. The planet's angular momentum is constant because the star's gravitational force is central.

* This constant angular momentum means the planet sweeps out equal areas in equal times, leading to a constant areal velocity.

Note: The areal velocity is a scalar quantity (has magnitude only) and is always positive.

This principle has important implications in understanding the motion of planets, satellites, and other objects moving under central forces.