NDA Physics · Formula sheet

Kinematics and Motion formulas

12 formulas, 1 reference table and 17 common traps for NDA Physics Kinematics and Motion, grouped by subtopic.

Foundations: Vectors, Distance, Displacement, and Position

Distance vs displacement

Distance and displacement

|\vec{s}| \leq d \qquad \vec{s}_{\text{round trip}} = 0

$d$ distance (total path length, scalar)
$\vec{s}$ displacement (start → finish, vector)

Speed, velocity, and their averages

Average speed and velocity

\bar{v}_{\text{speed}} = \dfrac{d}{t} \qquad \bar{v}_{\text{velocity}} = \dfrac{\vec{s}}{t}

$d$ total distance
$\vec{s}$ total displacement
$t$ total time

Position vector r(t) and net displacement

Position vector and its derivatives

\vec{v} = \dfrac{d\vec{r}}{dt}, \quad \vec{a} = \dfrac{d\vec{v}}{dt}, \quad |\vec{s}_{\text{net}}| = \sqrt{s_x^2 + s_y^2}

$\vec{r}$ position vector
$\vec{v}$ velocity vector
$\vec{a}$ acceleration vector

Scalars vs vectors

Quantity	Type	Why
Distance	Scalar	Total path length — no direction
Displacement	Vector	Straight-line change in position, with direction
Speed	Scalar	Rate of distance — magnitude onlyQ NDA 2022 — speed is scalar, velocity is vector. The single most-tested line of this subtopic.
Velocity	Vector	Rate of displacement — has direction
Acceleration	Vector	Rate of change of velocity — has direction

Pair each scalar with its vector cousin: distance/displacement, speed/velocity. The vector member always carries a direction.

Common traps

Speed is the scalar; velocity is the vector

The dominant distractor swaps them or calls both vectors. Speed is the magnitude of velocity (no direction); velocity carries a direction. A change in direction alone changes the velocity even when the speed is unchanged.

Round trip: distance is non-zero, displacement is zero

On any out-and-back trip the displacement is 0, so the average VELOCITY is 0 — but the distance and the average SPEED are not. The 50 km out-and-back PYQ tests exactly this: average velocity = 0, average speed = total distance / time.

Average speed is NOT |average velocity| in general

Average speed uses total distance; average velocity uses displacement. They match only for straight-line, no-reversal motion. Whenever a path curves or reverses, average speed > |average velocity|.

Add perpendicular legs as vectors, not as numbers

A common error sums the two leg distances arithmetically (100 + 200√2 ≈ 383 m). Perpendicular displacements must be combined with Pythagoras: √(100² + (200√2)²) = 300 m. Only collinear legs add directly.

Force ∥ momentum needs both vectors checked

For

\vec{r} = \sqrt{3}t^2\hat{i} + \sqrt{2}t\hat{j} + \sqrt{5}\hat{k}

, the force is

m\vec{a} = 2\sqrt{3}m\,\hat{i}

(pure

\hat{i}

) while at

t=0

the momentum is

m\sqrt{2}\,\hat{j}

(pure

\hat{j}

). They are perpendicular at that instant — differentiate twice and compare directions, do not assume force is along motion.

Equations of Motion and Motion Graphs

Learn this subtopic in the notes

Acceleration — rate of change of velocity

Acceleration

a = \dfrac{v - u}{t}

$a$ acceleration (m/s²)
$u$ initial velocity
$v$ final velocity
$t$ time interval

The three equations of motion

Equations of motion (constant a)

v = u + at \qquad s = ut + \tfrac{1}{2}at^2 \qquad v^2 = u^2 + 2as

$u$ initial velocity
$v$ final velocity
$a$ acceleration (constant)
$s$ displacement
$t$ time

Distance covered in the nth second

Distance in the nth second

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

$s_n$ distance during the nth second
$u$ initial velocity
$a$ acceleration
$n$ the second of interest

Reading a velocity-time graph

Velocity-time graph readings

a = \text{slope} = \dfrac{\Delta v}{\Delta t}, \qquad s = \text{area under the graph}

$\Delta v$ change in velocity
$\Delta t$ time interval
$s$ displacement

Reading a position-time graph

Position-time slope

v = \dfrac{dx}{dt} = \text{slope of the } x\text{-}t \text{ graph}

$x$ position
$t$ time
$v$ velocity (slope)

Common traps

Deceleration is negative acceleration, not 'no' acceleration

Slowing down is still acceleration — just opposite to the velocity, so it carries a minus sign in the direction of motion. From a velocity-time graph, average acceleration over an interval = (change in velocity) / (time), e.g. (4 − 8)/(12 − 8) = −1 m/s².

v² − u² = 2as, with the right sign

The third equation is

v^2 - u^2 = 2as

, i.e.

v^2 = u^2 + 2as

. A common wrong form writes

u^2 - v^2 = 2as

— that flips the sign and is the false option the bank tests. When decelerating, keep

a

negative rather than reordering

u

and

v

These equations need CONSTANT acceleration

v = u + at and friends apply only while a is constant. For a journey in two phases with different accelerations, apply the equations to each phase separately and add the results (as in the 'first t s at 2 m/s², next 10 s at 5 m/s²' problem).

The nth-second distance is not the total distance

sₙ = u + ½a(2n−1) gives the distance in a 1-second slice, not the cumulative s = ut + ½at². The valid NDA form is u + ½a(2n−1); options that drop the ½ or write (2n+1) are wrong.

Slope is acceleration; AREA is displacement — don't swap them

On a velocity-time graph the slope gives acceleration and the area gives displacement. A frequent error reads the area as a velocity or the slope as a distance. Also: a segment with positive slope (CD) is the accelerated part, a negative-slope segment (AB) is decelerated.

Check which axis is which before reading the slope

On the usual x-t graph the slope is the velocity directly. But if the figure plots TIME on the vertical axis against position, then speed = 1/slope, so the steepest line is the SLOWEST object — the reverse of the instinct.

Quadratic, not linear, distance growth

Under constant acceleration the distance grows as t² (a parabola), so 'distance increases linearly with time' is false. And it depends on the initial velocity u, not on any starting position.

Projectile and Vertical Motion

Learn this subtopic in the notes

Vertical throw — straight up under gravity

Vertical throw (up positive, a = −g)

v = u - gt, \qquad h_{\max} = \dfrac{u^2}{2g}, \qquad t_{\text{up}} = \dfrac{u}{g}

$u$ launch speed (upward)
$g$ acceleration due to gravity (≈ 10 m/s²)
$h_{\max}$ maximum height

Horizontal projectile — independence of motions

Horizontal projectile

t = \sqrt{\dfrac{2h}{g}}, \qquad R = u\,t

$u$ horizontal launch speed
$h$ launch height
$t$ time of flight
$R$ horizontal range

Common traps

Velocity is zero at the top, acceleration is NOT

At the highest point the velocity is momentarily 0, but the acceleration is still g downward — gravity never switches off. Use v = 0 only for the velocity, never set a = 0 at the top.

Horizontal speed never affects the fall time

A faster horizontal throw lands farther away but takes the SAME time to fall, because the vertical motion is independent free fall. Get the time from the height (t = √(2h/g)) first, then multiply by u for the range.

Circular Motion

Learn this subtopic in the notes

Centripetal acceleration — v²/r toward the centre

Centripetal acceleration

a_c = \dfrac{v^2}{r}

$a_c$ centripetal acceleration (toward centre)
$v$ speed
$r$ radius of the circular path

Average acceleration over part of a circle

Average acceleration over half a circle

\bar{a} = \dfrac{|\Delta \vec{v}|}{\Delta t} = \dfrac{2v}{\pi r / v} = \dfrac{2v^2}{\pi r}

$\Delta \vec{v}$ vector change in velocity (= 2v over half a circle)
$\Delta t$ time for the half circle (= πr/v)
$v$ constant speed

Common traps

Constant speed is not constant velocity

The single most common error in circular motion: treating 'uniform speed' as 'no acceleration'. The direction of velocity changes every instant, so the velocity is changing and there IS an acceleration — directed toward the centre.

Centripetal acceleration does not change the speed

The centripetal acceleration is perpendicular to the velocity (it points inward, velocity is tangent), so it changes only the DIRECTION of motion, never the speed. An option claiming 'centripetal acceleration causes the object to slow down' is false.

Average acceleration is not the instantaneous v²/r

v²/r is the instantaneous centripetal acceleration. The AVERAGE over half a circle uses |Δv|/Δt = 2v ÷ (πr/v) = 2v²/πr — a different expression. And over a full revolution the average acceleration is zero (Δv = 0), even though the instantaneous value is never zero.

Kinematics and Motion formulas

Foundations: Vectors, Distance, Displacement, and Position

Distance vs displacement

Speed, velocity, and their averages

Position vector r(t) and net displacement

Scalars vs vectors

Common traps

Equations of Motion and Motion Graphs

Acceleration — rate of change of velocity

The three equations of motion

Distance covered in the nth second

Reading a velocity-time graph

Reading a position-time graph

Common traps

Projectile and Vertical Motion

Vertical throw — straight up under gravity

Horizontal projectile — independence of motions

Common traps

Circular Motion

Centripetal acceleration — v²/r toward the centre

Average acceleration over part of a circle

Common traps

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