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Kinematics 2-D formulas

Master Kinematics 2-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 2-D, every formula

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

Horizontal velocity component
$u_{x} = u cos θ$
Q1MCQComponents
For a projectile launched at speed $u$ and angle $\theta$, the horizontal velocity is:
- A
  $u\cos\theta$
- B
  $u\sin\theta$
- C
  $u\tan\theta$
- D
  $u$
Vertical velocity component
$u_{y} = u sin θ$
Q1MCQVertical component
The initial vertical velocity of a projectile is:
- A
  $u\sin\theta$
- B
  $u\cos\theta$
- C
  $u$
- D
  $g t$
Time of flight (ground)
$T = \frac{2 u sin θ}{g}$
Maximum height
$H = \frac{u ^{2} sin ^{2} θ}{2 g}$
Horizontal range
$R = \frac{u ^{2} sin 2 θ}{g}$
Maximum range
$R_{m a x} = \frac{u ^{2}}{g} at θ = 4 5^{\circ}$
Q1MCQMax range angle
The range of a projectile is maximum at launch angle:
- A
  $45^{\circ}$
- B
  $30^{\circ}$
- C
  $60^{\circ}$
- D
  $90^{\circ}$
Trajectory equation
$y = x tan θ - \frac{g x ^{2}}{2 u ^{2} cos ^{2} θ}$
Q1MCQTrajectory
The path of a projectile is a:
- A
  parabola
- B
  straight line
- C
  circle
- D
  hyperbola
Velocity at time t
$v = u cos θ \hat{i} + (u sin θ - g t) \hat{j}$
Q1MCQVelocity vector
The velocity of a projectile at time $t$ is:
- A
  $u\cos\theta\,\hat i+(u\sin\theta-gt)\hat j$
- B
  $u\cos\theta\,\hat i+u\sin\theta\,\hat j$
- C
  $(u-gt)\hat i$
- D
  $u\hat i-gt\hat j$
Direction of velocity
$tan ϕ = \frac{u sin θ - g t}{u cos θ}$
Q1MCQVelocity direction
The angle of the velocity with the horizontal at time $t$ satisfies:
- A
  $\tan\phi=\dfrac{u\sin\theta-gt}{u\cos\theta}$
- B
  $\tan\phi=\dfrac{gt}{u}$
- C
  $\tan\phi=\tan\theta$
- D
  $\tan\phi=\dfrac{u\cos\theta}{u\sin\theta}$
Speed at highest point
$v = u cos θ$
v_y=0 there
Q1MCQHighest point speed
At the highest point of its path, a projectile's speed is:
- A
  $u\cos\theta$
- B
  $0$
- C
  $u$
- D
  $u\sin\theta$
Range–height relation
$R = 4 H cot θ$
Q1MCQRange–height
For a projectile, $R$ equals:
- A
  $4H\cot\theta$
- B
  $2H\tan\theta$
- C
  $4H\tan\theta$
- D
  $H\cot\theta$
Complementary angles
$θ and 9 0^{\circ} - θ give equal range$
Q1MCQComplementary angles
Two projectiles with the same speed at $30^{\circ}$ and $60^{\circ}$ have:
- A
  equal ranges
- B
  equal heights
- C
  equal times of flight
- D
  equal everything
Two times to reach height h
$h = \frac{g t _{1} t _{2}}{2}$
Q1MCQTwo times to height
If $t_1,t_2$ are the times to reach height $h$, then $h$ equals:
- A
  $\dfrac{g\,t_1 t_2}{2}$
- B
  $g\,t_1 t_2$
- C
  $\dfrac{g(t_1+t_2)}{2}$
- D
  $\dfrac{g\,t_1 t_2}{4}$
Horizontal projection: time
$T = \frac{2 H}{g}$
Horizontal projection: range
$R = u \frac{2 H}{g}$
Horizontal projection: path
$y = - \frac{g x ^{2}}{2 u ^{2}}$
Q1MCQHorizontal projection path
For a horizontally-projected body, the trajectory is:
- A
  $y=-\dfrac{gx^{2}}{2u^{2}}$
- B
  $y=-gx^{2}$
- C
  $y=ux$
- D
  $y=x\tan\theta$
Horizontal projection: angle
$tan ϕ = \frac{g t}{u}$
Q1MCQHorizontal projection angle
For a horizontally-projected body, the velocity angle satisfies:
- A
  $\tan\phi=\dfrac{gt}{u}$
- B
  $\tan\phi=\dfrac{u}{gt}$
- C
  $\tan\phi=gt$
- D
  $\tan\phi=u$
Projectile speed
$v = v_{x}^{2} + v_{y}^{2}$
Q1MCQProjectile speed
The speed of a projectile at any instant is:
- A
  $\sqrt{v_x^{2}+v_y^{2}}$
- B
  $v_x+v_y$
- C
  $v_x$
- D
  $v_y$
Time perpendicular to u
$t = \frac{u}{g sin θ}$
Q1MCQTime perpendicular
The time when the projectile's velocity is perpendicular to $\vec u$ is:
- A
  $\dfrac{u}{g\sin\theta}$
- B
  $\dfrac{u}{g\cos\theta}$
- C
  $\dfrac{2u\sin\theta}{g}$
- D
  $\dfrac{u}{g}$
Incline range (up)
$R = \frac{2 u ^{2} sin α cos ( α + β )}{g cos ^{2} β}$
Q1MCQIncline range up
The range up an incline is:
- A
  $\dfrac{2u^{2}\sin\alpha\cos(\alpha+\beta)}{g\cos^{2}\beta}$
- B
  $\dfrac{u^{2}\sin2\alpha}{g}$
- C
  $\dfrac{2u^{2}\sin\alpha\cos(\alpha-\beta)}{g\cos^{2}\beta}$
- D
  $\dfrac{u^{2}}{g}$
Incline range (down)
$R = \frac{2 u ^{2} sin α cos ( α - β )}{g cos ^{2} β}$
Q1MCQIncline range down
The range down an incline uses:
- A
  $\cos(\alpha-\beta)$
- B
  $\cos(\alpha+\beta)$
- C
  $\sin(\alpha+\beta)$
- D
  $\tan\beta$
Incline time of flight
$T = \frac{2 u sin α}{g cos β}$
Q1MCQIncline time
The time of flight on an incline is:
- A
  $\dfrac{2u\sin\alpha}{g\cos\beta}$
- B
  $\dfrac{2u\sin\alpha}{g}$
- C
  $\dfrac{2u\cos\alpha}{g\cos\beta}$
- D
  $\dfrac{u\sin\alpha}{g}$
Incline max range (up)
$R_{m a x} = \frac{u ^{2}}{g ( 1 + sin β )}$
Q1MCQIncline max range
The maximum range up an incline of angle $\beta$ is:
- A
  $\dfrac{u^{2}}{g(1+\sin\beta)}$
- B
  $\dfrac{u^{2}}{g(1-\sin\beta)}$
- C
  $\dfrac{u^{2}}{g}$
- D
  $\dfrac{u^{2}}{g\cos\beta}$
Incline max-range angle
$\frac{π}{4} - \frac{β}{2} (up), \frac{π}{4} + \frac{β}{2} (down)$
Q1MCQIncline max-range angle
For maximum range UP an incline, the projection angle with the incline is:
- A
  $\dfrac{\pi}{4}-\dfrac{\beta}{2}$
- B
  $\dfrac{\pi}{4}+\dfrac{\beta}{2}$
- C
  $\dfrac{\pi}{4}$
- D
  $\dfrac{\beta}{2}$
Relative velocity
$v_{A B} = v_{A} - v_{B}$
Relative acceleration
$a_{A B} = a_{A} - a_{B}$
Q1MCQRelative acceleration
The acceleration of $A$ relative to $B$ is:
- A
  $\vec a_A-\vec a_B$
- B
  $\vec a_A+\vec a_B$
- C
  $\vec a_B-\vec a_A$
- D
  $0$
River shortest time (net speed)
$v_{m} = v_{m R}^{2} + v_{R}^{2}$
Q1MCQRiver shortest time
To cross a river in the shortest time, a swimmer heads:
- A
  straight across (perpendicular to the banks)
- B
  upstream
- C
  downstream
- D
  at $45^{\circ}$
River shortest path (angle)
$θ = sin^{- 1} (\frac{v _{R}}{v _{m R}})$
Q1MCQRiver shortest path
For the shortest path across a river, the heading angle upstream is:
- A
  $\sin^{-1}\!\left(\dfrac{v_R}{v_{mR}}\right)$
- B
  $\sin^{-1}\!\left(\dfrac{v_{mR}}{v_R}\right)$
- C
  $0$
- D
  $45^{\circ}$
River shortest path (time)
$t = \frac{d}{v _{m R}^{2} - v _{R}^{2}}$
Q1MCQRiver crossing time
The time to cross by the shortest path is:
- A
  $\dfrac{d}{\sqrt{v_{mR}^{2}-v_R^{2}}}$
- B
  $\dfrac{d}{v_{mR}}$
- C
  $\dfrac{d}{v_R}$
- D
  $\dfrac{d}{\sqrt{v_{mR}^{2}+v_R^{2}}}$
Rain–man velocity
$v_{R m} = v_{R} - v_{m}, v_{R m} = v_{R}^{2} + v_{m}^{2}$

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Kinematics 2-D formulas

Kinematics 2-D, every formula

Horizontal velocity component

Vertical velocity component

Time of flight (ground)

Maximum height

Horizontal range

Maximum range

Trajectory equation

Velocity at time t

Direction of velocity

Speed at highest point

Range–height relation

Complementary angles

Two times to reach height h

Horizontal projection: time

Horizontal projection: range

Horizontal projection: path

Horizontal projection: angle

Projectile speed

Time perpendicular to u

Incline range (up)

Incline range (down)

Incline time of flight

Incline max range (up)

Incline max-range angle

Relative velocity

Relative acceleration

River shortest time (net speed)

River shortest path (angle)

River shortest path (time)

Rain–man velocity

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