Describing Motion Around Us
The kinematic equations
Use the three equations of motion to solve problems.
Why equations? When the acceleration stays constant, three equations let us predict where an object will be and how fast it will be going. They link five quantities:
- u = starting velocity;
- v = final velocity;
- a = acceleration;
- t = time;
- s = displacement.
You don't need to memorise them blindly: each one is built from an idea you already know.
Equation 1: the velocity after some time. Acceleration = change in velocity ÷ time, so a = (v – u) ÷ t. Multiply both sides by t and move u across:
v = u + at. (Memory hook: "VUAT".)
In words: final velocity = starting velocity + what you gained each second × the number of seconds. Example: start at 2 m/s, gain 3 m/s every second for 4 s: v = 2 + 3 × 4 = 14 m/s.
Equation 2: how far you go. The displacement is the area under the velocity-time line. That area is a rectangle (u × t) plus a triangle on top (½ × t × (v – u)). Since v – u = at, the triangle is ½ × t × at. So:
s = ut + ½at².
In words: the distance you would go at the starting speed, plus the extra from speeding up.
Equation 3: no time needed. Take t = (v – u) ÷ a from Equation 1 and put it into Equation 2. The t disappears, leaving:
v² = u² + 2as.
Use it when the question gives no time and asks for none.
How to choose the right one.
- Write down what you know.
- Write down what you want.
- Pick the equation that has exactly those letters.
Example: a car starts from rest (u = 0) and reaches 24 m s⁻¹ in 6 s. Want a: use v = u + at, so a = 24 ÷ 6 = 4 m s⁻². Want s: use s = ut + ½at² = ½ × 4 × 6² = 72 m.
Two warnings.
- The equations work only when the acceleration is constant.
- On a line, the signs (+ or –) of u, v, a and s show the directions. Slowing down means a is negative.
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