The equations of motion describe the relationship between the fundamental parameters of kinematics: distance, displacement, initial velocity, final velocity, acceleration, and time. These equations serve as a foundational tool in classical mechanics to predict the future state of a moving body or reconstruct its past trajectory.
Scope and Prerequisites for Derivation
To apply the standard equations of motion accurately, the physical system must fulfill specific conditions:
- Straight-Line Motion (Rectilinear Motion): The object must move along a single straight path in a one-dimensional coordinate system.
- Constant Acceleration: The acceleration must be uniform (a = constant). If acceleration varies with time or position, these algebraic equations fail, and calculus-based methods must be utilized instead.
- Frame of Reference: The motion must be observed from a consistent inertial frame of reference.
The Three Core Equations of Motion
The motion of an object experiencing uniform acceleration is mathematically defined by three distinct interconnected formulas.
First Equation of Motion: Velocity-Time Relation
This equation predicts the final velocity of an object after a specific time interval has elapsed under constant acceleration.
- u: Initial velocity (m/s)
- v: Final velocity (m/s)
- a: Uniform acceleration (m/s2)
- t: Time taken (s)
Second Equation of Motion: Position-Time Relation
This equation determines the total displacement covered by a moving body during a specific time interval.
- s: Displacement of the object (m)
Third Equation of Motion: Position-Velocity Relation
This equation correlates the initial and final velocities with displacement, completely independent of the time elapsed during the motion.
Special Case: Distance Covered in the nth Second
A critical variation derived from the second equation allows for the calculation of displacement specifically within a single chosen second (e.g., the 5th second of motion), rather than the cumulative displacement up to that point.
Derivations and Graphical Representations
In physics, these equations are visually and mathematically validated through velocity-time (v-t) graphs. For a uniformly accelerating object, the v-t graph is a straight line with a constant, non-zero slope.
Graphical Intercepts and Physical Properties
- Slope of a Velocity-Time Graph: Represents the acceleration (a) of the body.
- Area Under a Velocity-Time Graph: Represents the total displacement (s) traveled by the body.
Derivation of the First Equation (v = u + at)
The slope of the straight line on a v-t graph equals acceleration:
Derivation of the Second Equation (s = ut + 1/2at2)
The area under the v-t graph can be split into two geometric shapes: a rectangle representing motion at initial velocity, and a triangle representing the velocity added by uniform acceleration.
Derivation of the Third Equation (v2 = u2 + 2as)
Alternatively, the total area under the v-t graph can be calculated using the formula for a trapezium:
Motion Under Gravity: Vertical Trajectories
When an object moves vertically near the Earth’s surface, it experiences a constant downward acceleration due to gravity, denoted as g (≈ 9.8 m/s2). The standard equations are modified by substituting displacement s with height h, and acceleration a with ± g.
Cartesian Sign Convention for Gravity
To avoid calculation errors, a standard coordinate system must be applied:
- Upward direction is treated as Positive (+).
- Downward direction is treated as Negative (-).
- Acceleration due to gravity (g) always acts downward, so its value is substituted as -g in standard vector formulations.
Equations for an Object Thrown Vertically Upward
Since the motion is upward and gravity acts downward, the velocity decreases over time until it reaches zero at the peak.
- v = u – gt
- h = ut – 1/2gt2
- v2 = u2 – 2gh
Equations for a Freely Falling Object
When dropped from rest, the initial velocity u = 0. Both the displacement and gravitational acceleration point downward.
- v = gt
- h = 1/2gt2
- v2 = 2gh
Key Derived Formulae for Vertical Projectiles
| Physical Parameter | Definition | Formula |
| Maximum Height (Hmax) | The highest vertical displacement achieved before velocity hits zero. | Hmax = u2/2g |
| Time of Ascent (ta) | The duration taken to reach the maximum height. | ta = u/g |
| Time of Flight (T) | Total time spent in the air (Time of Ascent + Time of Descent). | T = 2u/g |
Core Trivia and Conceptual Facts for Prelims
Mass Independence in a Vacuum
The equations of motion under gravity do not include a mass variable (m). In 1971, Apollo 15 astronaut David Scott dropped a heavy hammer and a light feather simultaneously on the Moon (which lacks an atmosphere). Both objects hit the lunar surface at the exact same instant, validating Galileo’s theory that all objects experience identical gravitational acceleration regardless of mass when air resistance is absent.
Stopping Distance of Vehicles
When brakes are applied, a vehicle undergoes deceleration (negative acceleration). The stopping distance (d) can be derived from the third equation (v = 0):
The Concept of “Jerk”
While acceleration is the rate of change of velocity, the rate of change of acceleration with respect to time is known as a jerk (j = da/dt). It is measured in m/s3 and is a key parameter used by engineers to evaluate passenger comfort in elevators and trains.
Non-Uniform Acceleration Alternative
When acceleration depends on time, displacement, or velocity, the algebraic equations of motion become invalid. In such scenarios, kinematics relies on calculus: