The night before Minor 2

Motion graphs — cheat sheet

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

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 Minor 2)
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

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.

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

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.

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.