Formula Sheet: Motion in a Plane (2D Kinematics)

Yashwant Parihar,

In this post, we will learn all the formulas of Class 11 Chapter 3 Motion in a Plane with the help of a formula cheat sheet. The Motion in a Plane formula sheet covers everything from Projectile Motion equations to Uniform Circular Motion and Vector Addition tricks. Whether you are preparing for Class 11 exams or competitive tests like JEE and NEET, these formulas and shortcuts will help you solve complex 2D motion problems with ease.

List of All Motion in a Plane Formulas

1. Vector Fundamentals

Position Vector:

r=xi^+yj^r = x\hat{i} + y\hat{j}

Displacement:

δr=(x2−x1)i^+(Y2−Y1)j^\delta r = (x_2 – x_1)\hat{i} + (Y_2 – Y_1) \hat{j}

Magnitude of Resultant Vector:

R=A2+B2+2ABcosθR = \sqrt{A^2 + B^2 + 2ABcos\theta}

Direction of Resultant (alpha):

tanα=BsinθA+Bcosθtan\:\alpha = \frac{Bsin\theta}{A + Bcos\theta}

Dot Product:

A.B=ABcosθ(ResultisScaler)A.B = ABcos\theta (Result\:is\;Scaler)

Cross Product:

|A×B|=ABSinθ|A\times{B}| = ABSin\theta

Average Accerleration:

aavg=ΔvΔt=vxi^+vyj^ta_{avg} = \frac{\Delta{v}}{\Delta{t}}= \frac{v_x\hat{i}+v_y\hat{j}}{t}

2. Projectile Motion (The Most Important Part)

Time of Flight (T):

T=2usinθgT= \frac{2usin\theta}{g}

Maximum Height(H):

H=u2sin2θ2gH = \frac{u^2sin^2\theta}{2g}

Horizontal Range (R):

R=u2sin2θgR = \frac{u^2sin2\theta}{g}

Trick: If the range is maximum, then Theta = 45 °

R=u2gR = \frac{u^2}{g}

Equation of Tragectory Path:

y=xtanθ−gx22u2cos2θy = xtan\theta-\frac{gx^2}{2u^2cos^2\theta}

Velocity at any time t:

v=(ucosθ)2+(usinθ−gt2)v = \sqrt{(ucos\theta)^2+(usin\theta-gt^2)}

Radius of Curvature at the top of the Projectile’s Path:

ρ=u2cosθg\rho = \frac{u^2cos^\theta}{g}

3. Horizontal Projectile (From a Height h)

Time to hit the Ground:

t=2hgt = \sqrt{\frac{2h}{g}}

Horizontal Range:

R=u2hgR=u\sqrt{\frac{2h}{g}}

Velocity on Impact:

v=u2+2ghv = \sqrt{u^2+2gh}

4. Uniform Circular Motion

Angular Displacement:

Δθ=Δsr\Delta\theta = \frac{\Delta{s}}{r}

Angular Velocity:

ω=2πT\omega = \frac{2\pi}{T}
ω=2πf\omega = 2\pi{f}

Relation Between Linear and Angular Velocity:

v=rωv = r\omega

Centripetal Acceleration:

ac=v2r=ω2ra_c = \frac{v^2}{r} = \omega^2r

Total Acceleration (non-uniform):

atotal=ac2+at2a_{total} = \sqrt{a_c^2 +a_t^2}

5. Relative Motion in 2D

General Formula:

VAB=VA−VBV_{AB} = V_A -V_{B}

Rain-Man Problem:

To protect himself from rain falling vertically at vr, a man walking at vm must hold his umbrella at an angle theta:

tanθ=Vmvrtan\theta=\frac{V_m}{v_r}

River Boat Problem:

To cross a river of width (w) straight across, the boat must head at an angle:

sinθ=VriverVboatsin\theta = \frac{V_{river}}{V_{boat}}

Shortest time to cross a River:

tmin=ωVboatt_{min} = \frac{\omega}{V_{boat}}

JEE JEE Physics

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