Equations of Motion

• Three equations of uniformly accelerated motion (constant acceleration a).
• v = u + at — final velocity after time t.
• s = ut + ½at² — displacement in time t.
• v² = u² + 2as — velocity–displacement relation (no time needed).
• When to use each equation; sign convention for acceleration.

1. Symbols — u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.
2. First equation — v = u + at (linear relation between v and t).
3. Second equation — s = ut + ½at² (from area under v–t graph or calculus at Class 9 level).
4. Third equation — v² = u² + 2as (eliminates time).
5. Condition — valid only for uniform (constant) acceleration along a straight line.
6. Sign convention — take direction of motion as positive; braking → negative a.
7. Free fall — u = 0, a ≈ g = 9.8 m/s² downward (NCERT uses g = 10 m/s² for simplicity).
8. Real world — car accelerating from rest; ball dropped from a building.

A car accelerates uniformly from rest at 2 m/s² for 5 s. Find final velocity and distance.

Given: u = 0, a = 2 m/s², t = 5 s
Step 1 — v = u + at = 0 + 2 × 5 = 10 m/s
Step 2 — s = ut + ½at² = 0 + ½ × 2 × 25 = 25 m
Step 3 — Check with v² = u² + 2as: 100 = 0 + 2 × 2 × 25 = 100 ✓
Step 4 — Units: m/s for v, m for s, m/s² for a.

• Using equations when acceleration is not constant (e.g. varying engine power).
• Misconception: u must always be zero (only for 'from rest' problems).
• Wrong sign for a when object slows down (a opposite to v).
• Using distance instead of displacement in v² = u² + 2as when path is curved.
• Forgetting to convert km/h to m/s before substituting.

• Write all three equations of motion.
• A train slows from 20 m/s to 10 m/s in 50 m. Find deceleration.
• A stone is dropped (u = 0). How far does it fall in 3 s? (g = 10 m/s²)
• Which equation needs no time variable?

Open the Practice tab for graded questions on Equations of Motion.