Rectilinear Motion Short Notes for JEE Main: Revision Notes PDF

Displacement, velocity and acceleration

Distance and displacement

8× asked, last 2026

|\Delta\vec r\,| \le \text{distance}

Displacement: straight line start to end, a vector. Distance: path length, never negative. Equal only for one-way straight motion.

Not the arc for displacement. Displacement is the straight chord between the end points. An extra full lap adds to the distance only; an extra half-lap also moves the end point to the diametrically opposite point.

Arc of a circle

5× asked, last 2025

\text{distance}=R\theta,\qquad |\Delta\vec r\,| = 2R\sin\frac{\theta}{2}

\theta

in radians, the angle the arc makes at the centre. Quarter circle:

R\sqrt2

; half:

2R

; full lap: 0.

R\theta

|\Delta\vec r\,|

\theta

Ostartend

Distance: the arc

R\theta

. Displacement: the chord

2R\sin\frac{\theta}{2}

Average speed and average velocity

6× asked, last 2026

\bar v=\frac{\text{total distance}}{\text{total time}},\qquad \vec v_{\text{avg}}=\frac{\Delta\vec r}{\Delta t}

Average speed

\ge

magnitude of average velocity; equal only if the body never turns back on a straight line.

Not distance ÷ time for average velocity. Use displacement. A body that comes back to its start has zero average velocity, however far it went.

Equal distances or equal times

9× asked, last 2025

\bar v_{\text{equal } x}=\frac{2v_1v_2}{v_1+v_2},\qquad \bar v_{\text{equal } t}=\frac{v_1+v_2}{2}

One-way straight motion, so average speed = average velocity. Harmonic mean for equal distances, plain mean for equal times.

$n$ equal distances: $\bar v=n/(1/v_1+1/v_2+\dots+1/v_n)$ . For three parts this is $3v_1v_2v_3/(v_1v_2+v_2v_3+v_3v_1)$ .
Not $(v_1+v_2)/2$ for equal distances, nor the plain mean of any phase speeds. More time is spent at the slower speed. Divide total distance by total time.
Quick method. Mixed trips: replace each equal-time pair by its plain mean first, then take the harmonic mean over the equal-distance parts. Check: $v_1=v_2=v$ must give $v$ .

Instantaneous velocity and acceleration

8× asked, last 2025

v=\frac{dx}{dt},\qquad a=\frac{dv}{dt}=\frac{d^2x}{dt^2}

Differentiate first, put in

t

last. In 2D or 3D differentiate each component; a constant component gives zero.

$v$	$a$	The body is
$+$	$+$	speeding up ( $va>0$ )
$-$	$-$	speeding up ( $va>0$ )
$+$	$-$	slowing down ( $va<0$ )
$-$	$+$	slowing down ( $va<0$ )

Speed $=\sqrt{v_x^2+v_y^2+v_z^2}$ ; a minus sign only gives direction, squares always add. Angle with an axis: $\tan\theta$ = other component ÷ component along that axis.
On a line: speeding up when $v$ and $a$ have the same sign ( $va>0$ ), slowing down when they are opposite ( $va<0$ ). The sign of $a$ alone does not decide it: $a<0$ with $v<0$ is speeding up.
Not $x/t$ . If $x(0)=0$ that is the average velocity from 0 to $t$ , not the velocity at $t$ . And not the coefficient in $v=kt$ : that is the acceleration. Differentiate, then put in $t$ .
Not only the obvious $t^2$ term: expand brackets like $(t-2)^2$ , since every hidden $t^2$ adds to $a$ . Keep signs when you put $t=0$ for the initial velocity.

Average acceleration

2× asked, last 2023

\vec a_{\text{avg}}=\frac{\vec v_2-\vec v_1}{\Delta t}

A vector change. If the body reverses (a bounce), the speeds add:

|\Delta v|=v_1+v_2

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Practise Rectilinear Motion

31 more concepts in 4 topics

Equations of motion (uniform acceleration), Motion graphs, Motion under gravity and Variable acceleration. Read the first part free. Sign in free with Google to read the rest and download the PDF.

Equations of motion (uniform acceleration)

Equations of motion9×
Stopping distance and time4×
Displacement in the nth second4×
Starting from rest7×
Accelerate, then brake to rest3×
Velocity at the midpoint1×
Constant acceleration in a plane2×

Motion graphs

Position–time graph4×
Velocity–time graph10×
Acceleration–time graph3×
Graphs for constant acceleration2×
Velocity–position graph3×
Graph of v² against x2×

Motion under gravity

Free fall from rest10×
Thrown straight up8×
Thrown from a tower6×
Up, down and dropped from one point2×
Released from a moving body3×
Drops from a tap3×
Two bodies in free fall4×
Bouncing ball3×
Constant air drag1×
Smooth incline2×
Motion in stages2×

Variable acceleration

Acceleration from v(x)4×
Integrating v(t) and a(t)3×
Time given as a function of x2×
Implicit x–t relations2×
Turning points and extreme velocity4×
Force depending on x or v2×
Retardation depending on speed

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Rectilinear Motion revision notes

Displacement, velocity and acceleration

31 more concepts in 4 topics

Equations of motion (uniform acceleration)

Motion graphs

Motion under gravity

Variable acceleration

Practise Rectilinear Motion next

Also free on PrepWiser

Downloading Rectilinear Motion revision notes (PDF)

Rectilinear Motion revision notes

1Displacement, velocity and acceleration6 concepts

31 more concepts in 4 topics

2Equations of motion (uniform acceleration)7 concepts

3Motion graphs6 concepts

4Motion under gravity11 concepts

5Variable acceleration7 concepts

Practise Rectilinear Motion next

Also free on PrepWiser

Displacement, velocity and acceleration

Equations of motion (uniform acceleration)

Motion graphs

Motion under gravity

Variable acceleration