Equations of Motion: Understanding the Dynamics of Velocity, Acceleration, and Distance in Linear Motion

When an object travels in a straight line and experiences uniform acceleration, it abides by certain relationships between its various motion attributes: velocity, acceleration, and the distance covered. These relationships are captured in a set of equations commonly referred to as the “equations of motion”.

Three Primary Equations of Motion:

• Velocity-Time Relation: v = u + at. This equation relates the final velocity v of the object to its initial velocity u, the acceleration a, and the time t of motion.
• Position-Time Relation: s = ut + ½ at2.  Here, s represents the distance or displacement covered by the object in time t. 
• It relates the distance traveled by the object to its initial velocity, the time of motion, and the uniform acceleration.

• Position-Velocity Relation: 2 as = v2 – u2 . This equation connects the object’s displacement to its initial and final velocities, devoid of a direct time reference. 
• It can be derived from the previous two equations by eliminating the time t factor.
• In the equations:
  • u stands for the initial velocity.
  • v denotes the final velocity after time t.
  • a is the uniform acceleration.
  • s is the distance or displacement covered during the time t.
• Framework: These equations provide a foundational framework for analyzing linear motion with uniform acceleration. 
• Graphical representation: Notably, they can be derived using graphical methods, illustrating the power of graphical representations in understanding and deriving physical relationships.

Understanding the Dynamics of Circular Motion: From Acceleration to Equations of Motion (Uniform Circular)

• Change in Speed: Acceleration is usually associated with a change in the speed of an object.
  • However, it’s crucial to understand that a change in the direction of an object’s motion, even if its speed remains constant, also constitutes acceleration.

• Simplifying Circular Motion : A runner sprinting around a track. On a rectangular track, if the athlete maintains a constant speed on each straight section (AB, BC, CD, DA), the only time his velocity changes is when he rounds a corner. 
  • Here, he alters his direction four times during a single lap.
  • Modifying Track: Now, let’s modify the track’s shape. Suppose it’s hexagonal. On such a track, the runner adjusts his direction six times in a complete loop. 
    • If we shift to an octagonal track, the turns become more frequent — the athlete changes his direction eight times in one loop.
  • Increasing Numbers: The intriguing part comes when we keep increasing the number of sides on our track. 
    • As the number of sides approaches infinity, each side becomes infinitesimally small, and our shape morphs into a circle. 

• • • • Example: For an athlete maintaining constant speed on a circular track, the only fluctuation in his velocity stems from the continuous shifts in direction. 
      • Accelerated Motion: This movement, despite the speed being unaltered, is termed as accelerated motion due to the constant changes in direction.
      • Example: For a circular path with a radius r, its circumference is given by 2πr. If our runner takes t seconds for one complete round on this track, his speed v is computed as:
        v = 2πr / t​

• Conclusion: Thus when an object maintains a consistent speed while traveling in a circular path, it is termed uniform circular motion,  and its dynamics can be further understood through the equations of motion, as exemplified in the relationship between speed, radius, and time.