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Derivation
Derivation of Three Equations of Motion (Classical Mechanics)
📘 Derivation of Three Equations of Motion
Uniformly accelerated straight line motion | For school and entrance exam preparation 🚀
🎯 Basic Idea
The three equations of motion are valid for uniform acceleration in a
straight line. They relate five quantities:
initial velocity, final velocity, acceleration, time, and displacement.Symbol Physical quantity Unit (SI) u Initial velocity m s−1 v Final velocity m s−1 a Uniform acceleration m s−2 t Time taken s (second) s Displacement m (metre) Note: All three equations are derived under two conditions: (i) acceleration is constant;
(ii) motion is along a straight line.1️⃣ First Equation of Motion v = u + at
First equation of motion: v = u + at
Derivation (from definition of acceleration)
1. By definition, acceleration is the rate of change of velocity:
a = (v − u) / t
2. Rearranging the above equation:
v − u = at
3. Therefore,
v = u + at
Interpretation: Final velocity = Initial velocity + (change in velocity due to acceleration in time t).
2️⃣ Second Equation of Motion s = ut + ½at²
Second equation of motion: s = ut + \(\frac{1}{2}\)at²
Derivation (using average velocity)
1. For uniform acceleration, the average velocity is:
vavg = (u + v) / 2
2. Displacement in time t is:
s = vavg × t = ((u + v)/2) × t
3. From the first equation of motion, we know:
v = u + at
4. Substitute v = u + at in the expression for s:
s = [(u + (u + at)) / 2] × t
s = [(2u + at) / 2] × t
5. Multiply t inside:
s = (2u t + at²) / 2
6. Separate the terms:
s = ut + (1/2)at²
Interpretation: Displacement = distance covered due to initial velocity (ut) +
extra distance due to acceleration (\(\frac{1}{2}at^2\)).3️⃣ Third Equation of Motion v² = u² + 2as
Third equation of motion: v² = u² + 2as
Derivation (eliminating time t)
1. From the first equation of motion:
v = u + at
⟹ at = v − u ⇒ t = (v − u) / a
2. From the second equation of motion:
s = ut + (1/2)at²
3. Substitute t = (v − u)/a in this equation:
s = u × (v − u)/a + (1/2)a × ( (v − u)/a )²
4. Simplify step-by-step:
s = u(v − u)/a + (1/2)a (v − u)² / a²
s = u(v − u)/a + (1/2)(v − u)² / a
5. Take 1/a common:
s = [u(v − u) + (1/2)(v − u)²] / a
6. Multiply both sides by a:
as = u(v − u) + (1/2)(v − u)²
7. Expand the right-hand side:
u(v − u) = uv − u²
(v − u)² = v² − 2uv + u²
So,
as = (uv − u²) + (1/2)(v² − 2uv + u²)
8. Distribute (1/2):
as = uv − u² + (1/2)v² − uv + (1/2)u²
9. Combine like terms:
uv − uv = 0
−u² + (1/2)u² = −(1/2)u²
Thus,
as = (1/2)v² − (1/2)u²
10. Multiply both sides by 2:
2as = v² − u²
11. Rearranging:
v² = u² + 2as
This equation is useful when time is not given. It directly relates velocities, acceleration, and displacement.
🧮 Interactive Equations of Motion Calculator
Use this simple tool to calculate final velocity (v) and displacement (s)
for uniformly accelerated motion using:- v = u + at
- s = ut + (1/2)at²
Results will appear here…
Enter u, a, and t, then click “Calculate v and s”.