CDS Physics · Formula sheet

Kinematics and Motion formulas

5 formulas, 1 reference table and 15 common traps for CDS Physics Kinematics and Motion, grouped by subtopic.

Motion in a Straight Line

Distance, displacement and average speed

Average speed

\bar{v} = \dfrac{\text{total distance}}{\text{total time}} \qquad \bar{v}_{\text{equal distances}} = \dfrac{2v_1v_2}{v_1 + v_2} \qquad \bar{v}_{\text{equal times}} = \dfrac{v_1 + v_2}{2}

v₁, v₂the speeds on the two parts of the trip

Equations of motion for uniform acceleration

Equations of motion

v = u + at \qquad s = ut + \tfrac{1}{2}at^{2} \qquad v^{2} = u^{2} + 2as \qquad s = \dfrac{u + v}{2}\,t \qquad s_n = u + \dfrac{a}{2}(2n - 1)

$u, v$ starting and final velocity
$a$ constant acceleration (negative when slowing down)
$s$ displacement in time t
sₙdisplacement in the n-th second alone

Reading motion graphs

Graph and shape	What it means
Distance-time, horizontal line	The body is at restQ CDS 2019 (II): for a body at rest, the time-distance graph is horizontal.
Distance-time, steeper part	Higher speed in that partQ CDS 2023 (II): three parts of one journey are ranked by the slope of each part.
Velocity-time, horizontal line	Constant velocity, zero acceleration
Velocity-time, slope	Acceleration
Velocity-time, area under the line	DisplacementQ CDS 2023 (I): the area under a velocity-time graph gives the net displacement.

Common traps

Half a lap gives the diameter, not half the circumference

Displacement is the straight line between start and finish. After half a lap that line is the diameter. Half the circumference is the distance walked, which is a different answer.

Average speed over equal distances is not the plain average

Over equal distances the slower part takes longer, so the average is pulled down:

\dfrac{2v_1v_2}{v_1 + v_2}

. The plain average

\dfrac{v_1 + v_2}{2}

is right only for equal times.

Count whole laps first

Divide the time by the time for one lap. A whole number of laps means zero displacement; a half lap left over means one diameter. Do this before any other working.

Change minutes into hours before dividing

If the speed is in km/h, the time must be in hours. Ten minutes is

\tfrac{1}{6}

h, not 10. Mixing the two gives an answer off by a factor of 60.

The n-th second is one second, not n seconds

The distance in the n-th second is

u + \tfrac{a}{2}(2n - 1)

, the gap between the distance after n seconds and after (n − 1) seconds. Using

\tfrac{1}{2}an^{2}

gives the total from the start instead.

Retardation enters with a minus sign

When a body slows down, put

a

in as a negative number in

v = u + at

and

v^{2} = u^{2} + 2as

, or use the stopping form

s = \dfrac{u^{2}}{2a}

with the retardation as a positive number. Do not do both.

Flat on one graph is not flat on the other

A horizontal distance-time line means rest. A horizontal velocity-time line means constant velocity. Check which quantity is on the vertical axis before you read the shape.

Area gives displacement, not average velocity

The area under a velocity-time graph is the displacement. Divide it by the time if you want the average velocity. With part of the graph below the axis, the area gives the net displacement, not the total distance.

Motion in a Plane

Learn this subtopic in the notes

Adding vectors and resolving them

Resultant of two vectors

R = \sqrt{A^{2} + B^{2} + 2AB\cos\theta} \qquad |A - B| \le R \le A + B \qquad \vec{v} = \dfrac{d\vec{r}}{dt}, \ \vec{a} = \dfrac{d\vec{v}}{dt}

$A, B$ sizes of the two vectors
$θ$ angle between them
$R$ size of the resultant

Free fall and horizontal launch

h = \tfrac{1}{2}gt^{2} \qquad t = \sqrt{\dfrac{2h}{g}} \qquad v = \sqrt{2gh} \qquad R = u\sqrt{\dfrac{2h}{g}}

$h$ height of the fall
$g$ acceleration due to gravity
$u$ horizontal launch speed
$R$ horizontal distance to the landing point

Uniform circular motion

Centripetal acceleration

a = \dfrac{v^{2}}{r} = \omega^{2}r \qquad v = \omega r \qquad T = \dfrac{2\pi r}{v}

$v$ speed along the circle
$r$ radius of the circle
$ω$ angular speed in radians per second
$T$ time for one round

Common traps

Sizes add only when the directions match

|\vec{A}| + |\vec{B}|

equals

|\vec{A} + \vec{B}|

only at 0°. At any other angle the resultant is smaller than the sum of the sizes, never larger.

Combine forces along one line first

When several forces act along two perpendicular lines, first add the forces on each line (same way: add; opposite ways: subtract). Only then join the two line totals with Pythagoras.

sin and cos swap with the angle you measure from

Measured from the x-axis, the x-part uses cos and the y-part sin. Measured from the z-axis (the polar angle), the z-part uses cos and the part in the xy-plane uses sin.

The launch speed does not change the time in the air

For a horizontal launch the time comes from the height alone,

t = \sqrt{2h/g}

. The horizontal speed changes where the body lands, not when.

Acceleration stays constant, speed does not

In free fall the acceleration

g

is the one thing that stays uniform. Speed, momentum and kinetic energy all grow as the body falls.

Uniform circular motion is accelerated

Constant speed does not mean zero acceleration. The direction of the velocity keeps changing, so there is always an acceleration of size

v^{2}/r

Constant size is not constant acceleration

The centripetal acceleration has a fixed size but its direction turns with the body. A vector whose direction changes is not constant.

Kinematics and Motion formulas

Motion in a Straight Line

Distance, displacement and average speed

Equations of motion for uniform acceleration

Reading motion graphs

Common traps

Motion in a Plane

Adding vectors and resolving them

Uniform circular motion

Common traps

More CDS Physics formula sheets