How to Derive the Three Equations of Motion (Graphical, Algebraic & Calculus Methods)?
When a body moves along a straight line with constant acceleration, a specific relationship can be established between its velocity, acceleration, and the distance it travels during a particular time interval. Understanding the derivation of these relationships, known as equations of motion, is crucial in Physics. These equations are foundational for concepts in kinematics, enabling you to solve a wide variety of problems related to moving objects.
Grasping how these formulas are derived not only helps in solving numerical questions but also in developing a strong conceptual base for advanced topics.
Mathematical Derivation of Equations of Motion
Equations of motion apply when a body moves in a straight line with uniform acceleration. The three key quantities involved are:
- u: Initial velocity
- v: Final velocity after time t
- a: Uniform acceleration
- s: Distance (displacement) covered in time t
First Equation of Motion (Relation between Velocity and Time)
The first equation of motion links final velocity (v), initial velocity (u), acceleration (a), and time (t).
It states that the change in velocity is proportional to the acceleration and time:
| Equation | Description |
|---|---|
| v = u + at | Final velocity after time t |
Derivation:
Acceleration is the rate of change of velocity.
a = (v − u) / t
Rearranging gives: v = u + at
Second Equation of Motion (Relation between Displacement and Time)
The second equation connects displacement (s) with initial velocity (u), time (t), and acceleration (a):
| Equation | Description |
|---|---|
| s = ut + (1/2)at2 | Displacement after time t |
Derivation:
Total displacement (s) is calculated as:
s = average velocity × time
Average velocity = (u + v)/2
Substitute v from the first equation:
Average velocity = (u + (u + at)) / 2 = u + (at/2)
Therefore, s = [u + (at/2)] × t = ut + (1/2)at2
Third Equation of Motion (Relation between Velocity and Displacement)
This equation relates final velocity directly to displacement and acceleration, making it useful when time is not known.
| Equation | Description |
|---|---|
| v2 = u2 + 2as | Relates velocity to displacement |
Derivation:
From first equation: t = (v − u) / a
Substitute in the second equation:
s = ut + (1/2)at2
Replace t to eliminate it:
s = u[(v−u)/a] + (1/2)a[(v−u)/a]2
Solve to obtain: v2 = u2 + 2as
Key Formulas and Applications
| Name | Formula | Use Case |
|---|---|---|
| First Equation | v = u + at | Find final velocity after time t |
| Second Equation | s = ut + (1/2)at2 | Find distance/displacement in time t |
| Third Equation | v2 = u2 + 2as | Find velocity after displacement s |
Step-by-Step Approach to Solving Problems
- Identify what is given (u, v, a, s, t) and what you need to find.
- Choose the appropriate equation of motion based on known and unknown variables.
- Substitute the values into the formula.
- Solve for the unknown and state the final answer with correct units.
Example Problems
Example 1:
A body starts from rest (u = 0) and accelerates at 5 m/s2 for 4 seconds.
Find the final velocity and distance covered.
- Final velocity: v = u + at = 0 + 5 × 4 = 20 m/s
- Displacement: s = ut + (1/2)at2 = 0 + 0.5 × 5 × 16 = 40 m
Example 2:
A car moves with initial velocity 10 m/s and halts after covering a distance of 50 m under uniform retardation. Find the acceleration and time taken.
- v = 0; u = 10 m/s; s = 50 m
- Apply third equation: 0 = 100 + 2a × 50 → 2a × 50 = -100 → a = -1 m/s2
- Time: v = u + at → 0 = 10 + (-1)×t → t = 10 s
Related Vedantu Resources for Deeper Learning
- Equations of Motion by Graphical Method
- Kinematics Equations
- Motion in a Straight Line
- Uniformly Accelerated Motion
Practice Questions
- If a body moves with an initial velocity of 5 m/s with an acceleration of 2 m/s2 for 3 seconds, find the final velocity and the displacement.
- Derive the second equation of motion using the definition of acceleration.
- An object moves from rest and covers a distance of 100 m in 5 seconds with constant acceleration. Find its acceleration.
Key Points to Remember
- Equations of motion are valid only for uniform acceleration in a straight line.
- Use signs correctly: acceleration can be negative if the body is slowing down.
- Check units (m/s for velocity, m for distance, s for time) in every calculation.
- Learn to identify which variables are given and which equation suits the problem best.
For more on motion, acceleration, velocity, and advanced problem-solving, explore related resources such as Distance and Displacement, Velocity, Acceleration, and Newton's Laws of Motion.
FAQs on Equations of Motion: Derivations, Formulas & Applications Explained
1. What are the three equations of motion?
The three equations of motion describe how velocity, displacement, time, and acceleration are related for an object moving with uniform acceleration:
1. First Equation: v = u + at
2. Second Equation: s = ut + (1/2)at2
3. Third Equation: v2 = u2 + 2as
Where: u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.
2. How is the first equation of motion derived by the graphical method?
The first equation of motion (v = u + at) is derived using a velocity-time graph:
- Draw a straight line showing uniform acceleration from velocity u to v over time t.
- The slope gives acceleration (a).
- The change in velocity (v - u) occurs in time t.
- By definition, acceleration (a) = (v - u)/t → v = u + at.
3. What is the derivation of the second equation of motion using calculus?
The second equation of motion (s = ut + (1/2)at2) can be derived as follows:
1. Start with a = dv/dt (acceleration is the derivative of velocity).
2. Integrate velocity: v = u + at.
3. Displacement (s) is the integral of velocity:
s = ∫u dt + ∫a t dt = ut + (1/2)at2.
Thus, s = ut + (1/2)at2.
4. How do you derive the third equation of motion?
The third equation (v2 = u2 + 2as) is derived by eliminating time (t):
- Start from v = u + at.
- From s = ut + (1/2)at2, solve for t and substitute.
- Rearranging gives v2 = u2 + 2as.
This equation links final velocity, initial velocity, acceleration, and displacement without using time.
5. What is uniform acceleration?
Uniform acceleration means an object's velocity changes by equal amounts in equal intervals of time.
- Acceleration (a) remains constant.
- Graphically, the velocity-time graph is a straight line.
- All equations of motion apply only for uniform acceleration.
6. How can you apply the equations of motion to solve numerical problems?
To solve problems:
1. Write all given quantities (u, v, a, t, s).
2. Decide which equation of motion suits the unknown.
3. Substitute values and solve.
4. Always check units and answer logic.
This systematic approach helps avoid mistakes in board and entrance exam questions.
7. When should the first, second, or third equation be used?
Choose equations based on known and unknown values:
- Use first equation (v = u + at) when time, initial, and final velocities are involved.
- Use second equation (s = ut + 1/2 at2) when displacement and time are needed.
- Use third equation (v2 = u2 + 2as) if time is not given.
Always relate the equation to the required variable.
8. What are common mistakes in using equations of motion?
Common mistakes include:
- Using wrong sign for acceleration (positive/negative).
- Assuming initial velocity (u) is always zero.
- Mixing up distance and displacement.
- Applying equations to non-uniform acceleration.
Always read the problem carefully and assign correct values to each variable.
9. Can displacement ever be negative?
Yes, displacement can be negative if the final position is behind the initial position.
- Displacement is a vector quantity (has direction).
- Distance, by contrast, is always positive as it measures total path length.
10. How do you derive the equations of motion using the calculus method?
By calculus:
1. Use a = dv/dt and integrate to get v = u + at.
2. s = ∫v dt gives s = ut + (1/2)at2.
3. For v2 = u2 + 2as, use chain rule: a = v dv/ds and integrate.
This matches CBSE/NCERT method and reinforces concepts for JEE/NEET.
11. Why are equations of motion important in Physics?
Equations of motion form the basis of kinematics.
- They help describe the movement of objects mathematically.
- Essential for solving problems in mechanics, projectile motion, and real-world scenarios.
- Understanding and application are crucial for exams and further Physics studies.
12. What examiner tips help in mastering the derivation of equations of motion?
Examiner tips:
- Practice each derivation stepwise, using correct units and reasoning.
- Draw clear graphs for graphical methods.
- Highlight variables and formulae neatly.
- Review solved examples to recognize equation usage.
This builds stronger retention and boosts problem-solving skills for 2025 exams.
