The night before the Major

Kinematics — cheat sheet

Shape describes the motion, slope gives the rate, area gives the displacement.

Lessons 1–5 — units, vectors, speed & acceleration

SI units (L1)

length m · mass kg · time s · temperature K

Convert first:

km → m ×1000 · cm → m ÷100 · g → kg ÷1000
min → s ×60 · h → s ×3600 · °C → K +273
km/h → m/s ÷3.6

Scalars, vectors & journeys (L2–3)

Scalar = size only: distance, speed, time, mass, temperature.

Vector = size + direction: displacement, velocity, acceleration, force.

Distance: add every leg. Displacement: opposite directions subtract; right angles $\Delta x=\sqrt{a^2+b^2}$ , + direction. Back at the start → 0.

Speed, velocity, acceleration (L4–5)

\text{speed}=\dfrac{\text{distance}}{\text{time}}\;\;(\text{m/s})

v=\dfrac{\text{displacement}}{\text{time}}\;\;(\text{m/s + direction})

a=\dfrac{v_f-v_i}{t}\;\;(\text{m/s}^2)

v_f=v_i+a\,t\qquad t=\dfrac{v_f-v_i}{a}

From rest: vᵢ = 0 · comes to a stop: v_f = 0 · always final − initial.

Mistakes that lose marks: no unit · no direction on a vector · not converting first · adding opposite displacements · adding right-angle sides (use Pythagoras) · swapping vᵢ and v_f · "fast = accelerating" (constant velocity → a = 0) · forgetting an echo is there and back (halve it).

① Read the y-axis first.

Every rule depends on which graph it is. Most lost marks start with treating a V–T graph like a D–T graph.

	Displacement–time	Velocity–time
y-axis	Displacement (m)	Velocity (m/s)
Horizontal line	At rest — v = 0	Constant velocity — a = 0 (moving!)
Straight line sloping up	Constant velocity, positive direction	Velocity increasing — positive acceleration
Straight line sloping down	Constant velocity, negative direction	Negative acceleration (not negative velocity!)
Curve	Velocity changing (accelerating)	Acceleration changing (not on the Major)
Gradient =	velocity $v = \Delta d/\Delta t$	acceleration $a = \Delta v/\Delta t$
Area under =	nothing useful	displacement (below the axis = negative)
At rest shows as	a horizontal section	the line on v = 0
Changes direction when	the gradient changes sign (peak / trough)	the line crosses the time axis

Equations & units

$v = \dfrac{\Delta d}{\Delta t} = \dfrac{d_2 - d_1}{t_2 - t_1}$ m/s
gradient of D–T
$a = \dfrac{\Delta v}{\Delta t} = \dfrac{v_2 - v_1}{t_2 - t_1}$ m/s²
gradient of V–T
$\Delta d = \text{signed area under V–T}$ m
above +, below −
$\text{distance} = \sum |\text{areas}|$ m
all areas positive
$v_{avg} = \dfrac{\text{total displacement}}{\text{total time}}$ m/s
include time at rest

Area shapes

Rectangle $A = b \times h$
Triangle $A = \tfrac{1}{2} \times b \times h$
Trapezium $A = \tfrac{1}{2}(h_1 + h_2) \times b$

Speeding up or slowing down?

Compare the signs of v and a.

+v, +a

speeding up

+v, −a

slowing down

−v, −a

speeding up

−v, +a

slowing down

Same signs → speeding up. Opposite signs → slowing down.

D–TGraph 3 — Autonomous Vehicle Test

Steeper means faster: −4.5 m/s is faster than +3 m/s. The sign is only the direction.

V–TGraph 2 — Train Motion

Displacement = 18 + 6 − 4 − 8 − 4 = +8 m. Distance = 18 + 6 + 4 + 8 + 4 = 40 m.

Full marks pattern — don't lose the halves

v = Δd / Δt = (d₂ − d₁) / (t₂ − t₁) = (8 − 0) / (4 − 0) = 8 / 4 → v = +2 m/s

−½ missing/wrong equation
−½ missing/wrong unit
−½ missing sign / direction

Top traps

Horizontal V–T line = at rest. On a velocity–time graph a horizontal line means constant velocity (a = 0). It is only at rest if the line is on v = 0.
Negative slope = negative velocity. On a V–T graph a negative slope means negative acceleration. The sign of the velocity comes from being above (+) or below (−) the time axis.
Negative velocity is 'smaller'. The sign only gives the direction. Compare the size (magnitude): −4.5 m/s is faster than +3 m/s.
Area below the axis counted as positive. For displacement, area below the time axis is negative. Only use magnitudes when finding total distance.
Distance = displacement. Distance adds up every part of the path. Displacement is just final − initial position, with a direction.
Instant of rest read as an interval. If a V–T line only touches or crosses v = 0, the object is at rest for an instant, not for a period of time.
Direction change at a V–T peak. On a V–T graph the direction changes where the line crosses the time axis (v changes sign), not at a peak.
Average velocity = average of velocities. Average velocity = total displacement ÷ total time (include the time spent at rest).
'Stopped' means 'changed direction'. Stopping is not reversing. The direction changes only when the gradient (D–T) or velocity (V–T) changes sign.

“Right is Right” sentences

From __ s to __ s the object moves in the positive / negative direction at a constant velocity of __ m/s.
From __ s to __ s the object is stationary at a displacement of __ m (v = 0).
From __ s to __ s the object speeds up / slows down from __ m/s to __ m/s with an acceleration of __ m/s².
The acceleration is __ m/s² because acceleration is the gradient of a velocity–time graph.