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Kinematics 1-D formulas

Master Kinematics 1-D through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Kinematics 1-D, every formula

30 formulas, typeset and free. Print it, or keep it open beside your practice.

Displacement
$Δ x = x_{f} - x_{i}$
change in position
Average velocity
$⟨ v ⟩ = \frac{Δ x}{Δ t}$
Average speed
$avg speed = \frac{total distance}{total time}$
Instantaneous velocity
$v = \frac{d x}{d t}$
Q1MCQInstantaneous velocity
Instantaneous velocity is defined as:
- A
  $\dfrac{dx}{dt}$
- B
  $\dfrac{\Delta x}{\Delta t}$
- C
  $\dfrac{dv}{dt}$
- D
  $x\cdot t$
Average acceleration
$⟨ a ⟩ = \frac{Δ v}{Δ t}$
Instantaneous acceleration
$a = \frac{d v}{d t}$
Q1MCQInstantaneous acceleration
Instantaneous acceleration is:
- A
  $\dfrac{dv}{dt}$
- B
  $\dfrac{dx}{dt}$
- C
  $\dfrac{\Delta x}{\Delta t}$
- D
  $vt$
First equation of motion
$v = u + a t$
Second equation of motion
$s = u t + \frac{1}{2} a t^{2}$
Third equation of motion
$v^{2} = u^{2} + 2 a s$
Average-velocity equation
$s = \frac{u + v}{2} t$
Distance in nth second
$s_{n} = u + \frac{a}{2} (2 n - 1)$
Acceleration as v·dv/dx
$a = v \frac{d v}{d x}$
Q1MCQa = v dv/dx
When acceleration is a function of position, $a$ equals:
- A
  $v\dfrac{dv}{dx}$
- B
  $\dfrac{dv}{dt}$
- C
  $\dfrac{dx}{dt}$
- D
  $\dfrac{d^{2}v}{dx^{2}}$
Free fall from rest (speed)
$v = g t$
u=0
Free fall from rest (distance)
$h = \frac{1}{2} g t^{2}$
Maximum height (vertical throw)
$H = \frac{u ^{2}}{2 g}$
Time of flight (vertical throw)
$T = \frac{2 u}{g}$
Time to fall from height h
$t = \frac{2 h}{g}$
Speed on reaching ground
$v = 2 g h$
Galileo's odd-number ratio
$s_{1} : s_{2} : s_{3} : \dots = 1 : 3 : 5 : \dots$
equal time intervals
Q1MCQGalileo ratio
For a freely falling body, distances in successive equal time intervals are in the ratio:
- A
  $1:3:5:\cdots$
- B
  $1:2:3:\cdots$
- C
  $1:4:9:\cdots$
- D
  $1:1:1:\cdots$
Slope of x–t graph
$\frac{d x}{d t} = v$
Q1MCQx–t slope
The slope of a position–time graph gives:
- A
  velocity
- B
  acceleration
- C
  displacement
- D
  distance
Slope of v–t graph
$\frac{d v}{d t} = a$
Q1MCQv–t slope
The slope of a velocity–time graph gives:
- A
  acceleration
- B
  velocity
- C
  displacement
- D
  jerk
Area under v–t graph
$\int v d t = displacement$
Q1MCQv–t area
The area under a velocity–time graph gives:
- A
  displacement
- B
  acceleration
- C
  velocity
- D
  speed
Area under a–t graph
$\int a d t = Δ v$
Q1MCQa–t area
The area under an acceleration–time graph gives:
- A
  change in velocity
- B
  displacement
- C
  velocity
- D
  distance
Slope of v²–x graph
$\frac{d ( v ^{2} )}{d x} = 2 a$
Q1MCQv²–x slope
The slope of a $v^{2}$ versus $x$ graph equals:
- A
  $2a$
- B
  $a$
- C
  $v$
- D
  $\dfrac{a}{2}$
Relative velocity (1-D)
$v_{A B} = v_{A} - v_{B}$
Three-object throw relation
$t_{3} = t_{1} t_{2}$
up, down, dropped from same height
Q1MCQThree-object throw
For three bodies from the same height (thrown up $t_1$, down $t_2$, dropped $t_3$), $t_3$ equals:
- A
  $\sqrt{t_1 t_2}$
- B
  $t_1+t_2$
- C
  $\dfrac{t_1+t_2}{2}$
- D
  $t_1-t_2$
Distance vs displacement
$distance \geq ∣ displacement ∣$
Q1MCQDistance vs displacement
For any motion, distance is:
- A
  $\ge|\text{displacement}|$
- B
  $\le|\text{displacement}|$
- C
  $=$ displacement always
- D
  $=0$
a = f(t) integration
$\int d v = \int a d t$
Q1MCQa=f(t)
When $a=f(t)$, velocity is found from:
- A
  $\int dv=\int a\,dt$
- B
  $\int v\,dv=\int a\,dx$
- C
  $v=at$
- D
  $a=v\dfrac{dv}{dx}$
a = f(x) integration
$\int v d v = \int a d x$
Q1MCQa=f(x)
When $a=f(x)$, one uses:
- A
  $\int v\,dv=\int a\,dx$
- B
  $\int dv=\int a\,dt$
- C
  $v=u+at$
- D
  $s=ut$
Speed–velocity relation
$∣ avg velocity ∣ \leq avg speed$
Q1MCQSpeed vs velocity
The magnitude of average velocity compared with average speed is:
- A
  $\le$
- B
  $\ge$
- C
  $=$ always
- D
  unrelated

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Kinematics 1-D formulas

Kinematics 1-D, every formula

Displacement

Average velocity

Average speed

Instantaneous velocity

Average acceleration

Instantaneous acceleration

First equation of motion

Second equation of motion

Third equation of motion

Average-velocity equation

Distance in nth second

Acceleration as v·dv/dx

Free fall from rest (speed)

Free fall from rest (distance)

Maximum height (vertical throw)

Time of flight (vertical throw)

Time to fall from height h

Speed on reaching ground

Galileo's odd-number ratio

Slope of x–t graph

Slope of v–t graph

Area under v–t graph

Area under a–t graph

Slope of v²–x graph

Relative velocity (1-D)

Three-object throw relation

Distance vs displacement

a = f(t) integration

a = f(x) integration

Speed–velocity relation

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