Kinematics

Why this matters

Kinematics is the description of motion: where something is, how fast it is going and how quickly that speed changes. It comes before any question about why things move, and it shows up everywhere in engineering and science. A civil engineer checks how far a car travels while braking, a mechanical engineer works out how long a conveyor takes to bring a part up to speed, and a physicist reconstructs a particle's path from detector hits. All of them start with the same few quantities and four equations.

The quantities

• Position x (metres, m): where the object is, measured from an origin you choose.
• Displacement Δx = x − x₀ (m): the change in position. It has a sign. Moving 5 m left and then 5 m right is a displacement of 0 m, even though you walked 10 m.
• Velocity v (m/s): the rate of change of position. Speed is its size; velocity also carries direction.
• Acceleration a (m/s²): the rate of change of velocity. An acceleration of 3 m/s² means the velocity grows by 3 m/s every second.
• Time t (s).

Pick a positive direction at the start of every problem and stick with it. Any quantity pointing the other way gets a minus sign. Most kinematics mistakes are sign mistakes.

The constant-acceleration equations

When the acceleration stays constant, four equations connect the five quantities v₀ (starting velocity), v (final velocity), a, t and Δx:

1.  v  = v₀ + a·t
2.  Δx = v₀·t + ½·a·t²
3.  v² = v₀² + 2·a·Δx
4.  Δx = ½·(v₀ + v)·t

Each equation leaves out one quantity. Equation 1 has no Δx, equation 2 has no v, equation 3 has no t and equation 4 has no a. The strategy is always the same: list the three values you know, name the one you want, and pick the equation that does not contain the fifth (the one you neither know nor need).

These equations are only valid while the acceleration is constant. If a problem has two phases (speeding up, then cruising), solve each phase separately and use the end of the first phase as the start of the second.

Worked example 1: a car pulling away

A car starts from rest and accelerates at 3.0 m/s² for 6.0 s. How fast is it going, and how far has it travelled?

Known: v₀ = 0, a = 3.0 m/s², t = 6.0 s.

v  = 0 + 3.0 × 6.0        = 18 m/s   (about 65 km/h)
Δx = 0 × 6.0 + ½ × 3.0 × 6.0² = 54 m

Check with equation 4: ½ × (0 + 18) × 6.0 = 54 m. Two equations agreeing is a quick way to catch arithmetic slips.

Worked example 2: stopping distance

A car travels at 25 m/s (90 km/h) and brakes with a constant deceleration of 7.0 m/s². How far does it slide before stopping?

Take the direction of travel as positive, so a = −7.0 m/s². Known: v₀ = 25 m/s, v = 0, a = −7.0 m/s². We do not know or need t, so use equation 3:

0 = 25² + 2 × (−7.0) × Δx
Δx = 625 / 14 = 44.6 m

In general the braking distance is v₀² / (2·|a|). Because speed is squared, doubling the speed makes the braking distance four times longer. That single fact explains a lot of road design.

Real stopping distance also includes the driver's reaction time. During a 1.5 s reaction the car still moves at 25 m/s, covering 37.5 m before the brakes even start, so the total is about 82 m. Road and vehicle standards use carefully chosen values for reaction time and friction; never take the numbers in a lesson as design values.

Worked example 3: free fall

Near Earth's surface, anything falling freely (ignoring air resistance) accelerates downward at g ≈ 9.81 m/s². Drop a ball from a 20 m balcony. Take down as positive, so a = +9.81 m/s², v₀ = 0, Δx = 20 m.

20 = ½ × 9.81 × t²   →   t = √(40 / 9.81) = 2.02 s
v  = 9.81 × 2.02 = 19.8 m/s

The same idea in code

Engineers rarely solve these by hand more than once. A function makes the equations reusable and easy to test:

def displacement(v0, a, t):
    """Displacement after t seconds of constant acceleration."""
    return v0 * t + 0.5 * a * t ** 2

print(displacement(0, 3.0, 6.0))   # 54.0

Common mistakes

• Mixing units. Convert km/h to m/s first: divide by 3.6. 90 km/h is 25 m/s.
• Forgetting signs. A car that is slowing down while moving forward has a negative acceleration in your chosen frame.
• Using the equations when a changes. If the force (and so the acceleration) changes, split the motion into phases or use a numerical method.
• Confusing distance and displacement. The equations give displacement.

Quick recap

• Choose a positive direction and give every vector quantity a sign.
• With constant acceleration, four equations link v₀, v, a, t and Δx. Pick the one that leaves out the quantity you do not need.
• Braking distance grows with the square of speed.
• Free fall near Earth: a = 9.81 m/s² downward.

Try it

Open the Do tab and write the stopping-distance function, then use it to see how braking distance changes with speed.