Equations of Motion
The equations of motion relate displacement, velocity, acceleration, and time for objects moving with constant acceleration. We use the following symbols:
v: Final Velocity
a: Constant Acceleration
t: Time
s: Displacement
Equation 1: The Velocity Equation
v = u + at
Derivation: Acceleration (a) is the rate of change of velocity: a = (v – u) / t. Rearranging gives v = u + at.
Equation 2: The Displacement Equation
s = ut + ½at²
Derivation: Displacement is the area under a v-t graph (rectangle + triangle): Area = ut + ½(t)(v – u). Since v – u = at, we get s = ut + ½at².
Equation 3: The Timeless Equation
v² = u² + 2as
Derivation: Combining v = u + at and s = [(u + v) / 2] * t to eliminate time (t) results in v² = u² + 2as.
Example Problem: The Braking Car
Problem: A car traveling at 30 m/s applies its brakes and stops in 6 seconds. Calculate acceleration and distance.
1. Givens: u = 30 m/s, v = 0 m/s, t = 6 s
2. Acceleration: 0 = 30 + a(6) → a = -5 m/s² (Deceleration)
3. Displacement: s = (30)(6) + ½(-5)(6²) → s = 90 meters