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Third Equation of Motion
The third equation connects initial velocity, final velocity, acceleration and displacement: v² = u² + 2as. It does not contain time.
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Third Equation of Motion
15. Third Equation
The third equation connects:
- Initial velocity
- Final velocity
- Acceleration
- Displacement
It does not contain time. v² = u² + 2as
16. Derivation of the Third Equation
Average velocity is: vavg = u + v2 Displacement is: s = (u + v2) t From the first equation: v = u + at Rearranging for time: v − u = at t = v − ua Substituting this value of t: s = (u + v2) × (v − ua) Using: (u + v)(v − u) = v² − u² Therefore: s = v² − u²2a Multiplying both sides by 2a: 2as = v² − u² Adding u² to both sides: v² = u² + 2as Therefore: v² = u² + 2as
17. Solved Example 3: Braking Distance
Solved Example 3: Braking Distance
A car moving at 20 m s⁻¹ comes to rest with a uniform acceleration of −5 m s⁻². Find its stopping distance.
- u = 20 m s⁻¹
- The car comes to rest: v = 0
- a = −5 m s⁻²
- Required: s = ?
- Formula: v² = u² + 2as
(0)² = (20)² + 2(−5)s
0 = 400 − 10s
10s = 400
s = 40 m
Key Step: The acceleration is negative because it acts opposite to the car’s velocity.
Final answer: s = 40 m
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Practice
Practice
Q1. A vehicle slows from 25 m s⁻¹ to 5 m s⁻¹ at −4 m s⁻². Find the distance travelled.
Practice
Q2. Derive the third equation by eliminating time.