Derive equations of motion for a particle under constant acceleration.

Short Answer:

When a particle moves with constant (uniform) acceleration, its velocity changes at a steady rate. Starting from initial velocity , after time  the velocity becomes . The particle’s displacement  in time  is given by . These simple algebraic relations let you find any one of  when the others are known.

These three standard equations — , , and  — are derived by integrating the definitions of velocity and acceleration (or by using average velocity). They assume motion in a straight line with acceleration constant in magnitude and direction.

Equations of motion for constant acceleration

Assumptions and notation

  • Particle moves along a straight line.
  • Acceleration  is constant (both magnitude and direction).
  •  = initial velocity at time .
  •  = velocity at time .
  •  = displacement in time  measured from initial position (so  at ).
  • Time measured from the instant when velocity = .

Derivation using calculus (clean, general way)

Acceleration is the rate of change of velocity:

Because  is constant we integrate with respect to time from  to . Rearranging,

Integrate both sides:  This gives

This is the first equation of motion.

Velocity is the rate of change of displacement:

Substitute  from the result above (or integrate directly using ):

Integrate from  to :  This gives

This is the second equation of motion.

To get a relation that does not involve time explicitly, start from  and write  because . Then

Rearrange and integrate between initial and final states:

This yields

This is the third equation of motion. It is useful when time is not known or not needed.

Derivation using average velocity (elementary algebraic way)

When acceleration is constant, average velocity over time interval  equals the arithmetic mean of initial and final velocities:

Displacement equals average velocity times time:

Using  to eliminate  gives

recovering the second equation. Eliminating  between  and  gives .

Notes on vector form and signs

  • The equations above are scalar forms along one straight line. If motion is on a line with a chosen positive direction, signs of  follow that choice. Negative acceleration simply means acceleration opposite to chosen positive direction (retardation if velocity and acceleration are opposite).
  • For two- or three-dimensional motion with constant vector acceleration , the same relations hold vectorially:  and . The scalar  uses dot product.

Practical remarks

  • These equations apply only while acceleration is constant. If  changes with time or position, you must integrate the actual  or use numerical methods.
  • Common simple examples: free fall near Earth’s surface (take  downward), uniformly accelerated car motion, motion on a smooth inclined plane with constant component of gravity.
Conclusion

For a particle with constant acceleration, three primary kinematic relations connect displacement, velocity, acceleration, and time: , , and . They follow directly from the definitions of velocity and acceleration by integration or from the average velocity idea. These equations form the foundation for solving many straight-line motion problems in mechanics when acceleration does not change.