Master Motion Formula Table

Complete reference for Linear (SUVAT), Projectile, Circular, and Rotational motion equations.

Linear

First Equation of Motion

v = u + at
Missing:Displacement (s)
Finds final velocity without distance.
Car accelerating from rest for 5s.
Linear

Second Equation of Motion

s = ut + ½at²
Missing:Final Velocity (v)
Finds displacement given time and acceleration.
Distance fallen by a dropped stone in 3s.
Linear

Third Equation of Motion

v² = u² + 2as
Missing:Time (t)
Timeless equation. Finds velocity or info without time.
Braking distance of a truck.
Linear

Average Velocity Formula

s = ½(u + v)t
Missing:Acceleration (a)
Finds displacement using average velocity.
Distance covered while smoothly slowing down.
Linear

Displacement from End

s = vt - ½at²
Missing:Initial Velocity (u)
Less common. Finds calc from the end state backwards.
Reconstructing start point given stop point.
Projectile

Time of Flight

T = (2u sin θ) / g
Total time the projectile stays in the air.
How long a football stays airborne.
Projectile

Maximum Height

H = (u² sin² θ) / 2g
The peak vertical position reached.
Highest point of a rocket launch.
Projectile

Horizontal Range

R = (u² sin 2θ) / g
Total horizontal distance covered.
How far a golf ball travels.
Projectile

Trajectory Equation

y = x tan θ - (gx²) / (2u² cos² θ)
The parabolic path equation y(x).
Plotting the curve of a water jet.
Circular

Angular Velocity

ω = 2πf or ω = v/r
Rate of rotation in radians per second.
Speed of a spinning fan blade.
Circular

Centripetal Acceleration

a_c = v²/r or a_c = ω²r
Acceleration directing towards the center.
Force felt on a turning carousel.
Circular

Centripetal Force

F_c = (mv²)/r
Net force keeping object in circle.
Tension in string swinging a ball.
Circular

Period of Revolution

T = 2πr / v
Time taken for one complete circle.
Time for Earth to orbit Sun.
Rotational

Rotational Kinematics (1)

ωf = ωi + αt
Analogue to v = u + at.
Wheel speeding up to final rpm.
Rotational

Rotational Kinematics (2)

θ = ωit + ½αt²
Analogue to s = ut + ½at².
Total angle turned during spin up.
Rotational

Torque

τ = Iα or τ = rF sin θ
Rotational force causing angular acceleration.
Effort to open a heavy door.
Rotational

Kinetic Energy (Rot)

KE = ½Iω²
Energy stored in a spinning object.
Energy in a localized flywheel.
Showing 17 formulasTip: Click equation to copy

The Language of Movement

Whether it's a car braking at a red light, a football soaring through the air, or the Earth spinning on its axis, everything follows the same set of mathematical rules. This tool compiles the essential equations of Kinematics (Motion) into one master reference.

Linear Motion (SUVAT)

These 5 equations describe objects moving in a straight line with constant acceleration. They are the bread and butter of high school physics. The key is to list your variables ($s, u, v, a, t$) and pick the equation that matches what you have.

Rotational Motion

Spinning objects follow the exact same logic as moving ones, just with different symbols. We replace meters with radians, and kg with Moment of Inertia ($I$). If you know the linear formulas, you automatically know the rotational ones by analogy!

The "Linear to Rotation" Dictionary

Physics is beautiful because of its symmetry. Use this table to translate linear concepts into rotational ones.

ConceptLinear SymbolRotational SymbolRelation
Displacement$s$ (meters)$\\theta$ (radians)$s = r\\theta$
Velocity$v$ (m/s)$\\omega$ (rad/s)$v = r\\omega$
Acceleration$a$ (m/s²)$\\alpha$ (rad/s²)$a = r\\alpha$
Inertia / Mass$m$ (kg)$I$ (kg·m²)$I = mk^2$
Force / Torque$F$ (Newtons)$\\tau$ (N·m)$\\tau = rF$

Solver Tip: The "No-Variable" Rule

Struggling to pick an equation? Look at what variable you are NOT given and NOT asked to find.
  • No Time ($t$)? Use $v^2 = u^2 + 2as$.
  • No Distance ($s$)? Use $v = u + at$.
  • No Final Vel ($v$)? Use $s = ut + 0.5at^2$.

Frequently Asked Questions

What are the 4 SUVAT equations?

The 4 key equations are: 1) $v = u + at$, 2) $s = ut + 0.5at^2$, 3) $v^2 = u^2 + 2as$, and 4) $s = 0.5(u+v)t$. They act as a toolkit to solve any constant acceleration problem.

When can I use SUVAT equations?

You can ONLY use them when acceleration is CONSTANT (uniform). If acceleration is changing (like a car with changing throttle), you must use Calculus. For gravity problems near Earth, acceleration ($g$) is constant, so SUVAT works perfectly.

How do I choose the right equation?

Identify the variable you are MISSING and don't care about. For example, if you don't know Time ($t$) and don't need to find it, use the equation $v^2 = u^2 + 2as$ because it has no $t$ in it.

What is the "Range" formula for projectiles?

The horizontal range $R = (u^2 \sin 2\theta) / g$. This assumes the projectile lands at the same height it was launched. Maximum range occurs at 45 degrees.

What is the difference between Linear and Rotational variables?

They are analogs! Linear Displacement ($s$) becomes Angle ($\theta$). Velocity ($v$) becomes Angular Velocity ($\omega$). Acceleration ($a$) becomes Angular Acceleration ($\alpha$). Mass ($m$) becomes Moment of Inertia ($I$). Force ($F$) becomes Torque ($\tau$).

Why is Centripetal Acceleration $v^2/r$?

Even if speed is constant in a circle, direction is always changing. This change in velocity direction requires an inward acceleration of $v^2/r$. Without it, the object would fly off in a straight line (tangent).

Does mass affect the time of flight?

In vacuum (no air resistance), NO. A feather and a hammer fall at the same rate. However, in real life with air resistance, mass and shape do matter (terminal velocity).

What is "Moment of Inertia" (I)?

It is the rotational equivalent of Mass. It measures how hard it is to start spinning an object. $I$ depends on mass AND how that mass is distributed (further from center = harder to spin).

What is the value of g?

On Earth, $g \approx 9.81 m/s^2$. In problems, we often use $g = 9.8$ or even $10 m/s^2$ for estimation. It always points DOWN towards the center of Earth.

How do I convert rpm to rad/s?

To convert Revolutions Per Minute (rpm) to Radians Per Second (rad/s): Multiply by $2\pi$ and divide by 60. Formula: $\omega = rpm \times (2\pi / 60) \approx rpm \times 0.1047$.