Question

Discuss the motion of the earth with special reference to distance covered, displacement, speed, velocity and acceleration

Answer 1

Earth moves in a plane in a circular path (well actually a slightly elliptical path) around the Sun.  The Sun is at one of the Foci of the ellipse.  In addition to this revolution, Earth has also a rotational motion around itself taking 24 hours for a rotation.

Let us assume that the Earth moves in a circle of Radius R with Sun at the center.  This radius is much larger than radius of Earth.  The speed of Earth can be assumed to be nearly constant.

Thus the angular speed of Earth is a constant.  This is equal to :  
     ω = 2π/(365.25 *86400)  rad/sec.
     R = 1.5 * 10¹¹ m  nearly.

The distance covered in time "t" will be
       = d = R (ω t)  meters (length of arc).

The displacement s will be the length of the chord connecting initial and final positions.  Obviously magnitude will be less than the distance covered.
Displacement direction also is along the chord, from initial to final position.

   displacement = s = 2 * R Sin (ω t/2)
   The displacement will have the maximum value of  2 R = diameter.

Speed = u = angular speed * radius = distance/time = R ω

Linear Velocity = v = ds/dt = R ω Cos (ωt/2)
The direction of velocity will be along the tangent to the circle.

Since there is no external force acting on the system of Sun and Earth (assumption), Earth moves with constant speed.  There is no tangential acceleration.  However as the direction of velocity changes, there is a radial acceleration (called centripetal acceleration).  The cause of this is the Gravitational attraction of Sun.

   Radial acceleration = v² / R = R ω²
This vector has a constant magnitude, but its direction keeps changing and is always along the radius connecting Earth and Sun.