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Equations of Motion (Constant Acceleration)
Velocity as a function of time under constant acceleration.
Interpretation
Equations of motion for constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as. These relate velocity, displacement, acceleration, and time. Example: u=0, a=2 m/s², t=5s → v=10 m/s.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Final velocity | m/s |
| u | Initial velocity | m/s |
| a | Acceleration | m/s² |
| t | Time | s |
What it means
The three equations of motion describe the relationship between displacement, initial velocity, final velocity, acceleration, and time for an object moving with constant acceleration. They are derived from the definitions of acceleration and average velocity. The first equation v = u + at gives the final velocity after time t; the second s = ut + ½at² gives the displacement; and the third v² = u² + 2as relates velocity and displacement without involving time. These equations are fundamental in kinematics and are used extensively in physics and engineering to analyse projectile motion, vehicle acceleration, and any uniformly accelerated motion. They assume constant acceleration and straight‑line motion. In practice, they are applied in designing braking systems, predicting motion of falling objects, and calculating trajectories. Understanding these equations is essential for solving a wide range of problems in mechanics.
Worked example
Equations of Motion – Two Examples
Real‑World| Parameter | Value |
|---|---|
| u | 0 m/s |
| a | 2 m/s² |
| t | 5 s |
| Parameter | Value |
|---|---|
| u | 5 m/s |
| a | 1.5 m/s² |
| t | 4 s |
Common mistakes
- Sign convention: v = u + at uses a sign convention for direction. Define positive direction consistently (e.g., upwards positive, downwards negative).
- Constant acceleration: This equation assumes constant acceleration. Do not use it for varying acceleration unless using calculus.
- Unit consistency: v and u in m/s, a in m/s², t in s – ensure all SI units.
- Initial velocity u: It is the velocity at t=0; if the object starts from rest, u=0.
- Final velocity v: The velocity after time t – not the displacement.
Applications
The equations of motion under constant acceleration – v = u + at, s = ut + ½at², and v² = u² + 2as – are the foundation of classical kinematics. They describe the relationship between displacement, velocity, acceleration, and time for objects moving with uniform acceleration. These equations are used extensively in engineering and physics to predict the motion of vehicles, projectiles, and machinery. In automotive design, they help calculate stopping distances and acceleration times. In sports science, they model the trajectory of balls and athletes. In aerospace, they are used for launch and landing trajectories. By applying these equations, engineers can design safety systems, optimise performance, and analyse dynamic systems. Their simplicity and power make them essential tools in both education and professional practice across all fields involving motion.
- Vehicle braking distance and acceleration calculations
- Projectile motion analysis in ballistics and sports
- Design of amusement park rides and elevators
- Launch and landing trajectory planning in aerospace
- Educational foundation for introductory physics and engineering
Frequently Asked Questions
For constant acceleration, we have four kinematic equations:
1) v = u + at (velocity after time t).
2) s = ut + ½at² (displacement after time t).
3) v² = u² + 2as (velocity‑displacement relation).
4) s = ½(u+v)t (displacement from average velocity).
They apply to any motion with constant acceleration (including free‑fall, braking, and projectile motion).
- u – initial velocity (m/s).
- v – final velocity (m/s).
- a – constant acceleration (m/s²).
- t – time interval (s).
- s – displacement (m).
Choose a positive direction (e.g., upward or right). Any vector pointing in that direction is positive; opposite is negative. For example, if upward is positive, the acceleration due to gravity is a = –g (≈ –9.81 m/s²). Consistent sign use is essential to get correct answers.
Projectile motion has constant horizontal velocity and constant vertical acceleration (g downward). Horizontally: use vx = ux and sx = ux·t. Vertically: use the full set with a = –g. The two motions are independent and are linked by time.
Average acceleration is the change in velocity divided by the time interval: a_avg = Δv/Δt. Instantaneous acceleration is the derivative a = dv/dt at a specific instant. The equations of motion assume constant (uniform) acceleration, so average and instantaneous are equal.
For variable acceleration, you cannot use these equations directly. You must integrate: v = ∫a·dt and s = ∫v·dt. If a is given as a function of time, velocity, or position, use differential equations. Numerically, you can also integrate using step‑by‑step methods.
List the known and unknown variables. Choose the equation that contains the unknown and only known quantities. For example, if you have u, v, a and want s, use v² = u² + 2as (time not needed). If you have u, a, t and want s, use s = ut + ½at².
- Using the wrong sign for acceleration.
- Forgetting that displacement, velocity, and acceleration are vectors – using magnitudes incorrectly.
- Mixing units (e.g., using km/h and m/s together).
- Assuming constant acceleration when it is not.
- Applying the equations to variable acceleration without integration.
They follow from the definitions: a = dv/dt → integrate to get v = u + at. Then v = ds/dt → integrate to get s = ut + ½at². The third is obtained by eliminating t: v² = u² + 2as.
- Vehicle braking distance calculations.
- Rocket trajectory analysis.
- Roller coaster design.
- Sports physics (ballistics).
- Elevator acceleration profiles.