Physics formula sheet
Every formula used in The Physics Lab in one place: kinematics, projectile motion, forces, energy, oscillations, waves, electricity, and optics. Each entry says what the formula finds and when it applies, and every group links to a simulation that shows it working.
Motion formulas
Projectile motion
- Horizontal velocity
- vₓ = v₀ cos θ
- Constant for the whole flight
- Initial vertical velocity
- vᵧ = v₀ sin θ
- Gravity reduces it by g each second
- Position at time t
- x = v₀ cos θ · t, y = v₀ sin θ · t − ½gt²
- The path is a parabola
- Time of flight
- T = 2v₀ sin θ / g
- Level launch and landing
- Maximum height
- H = v₀² sin²θ / (2g)
- Reached halfway through the flight
- Range
- R = v₀² sin 2θ / g
- Largest at θ = 45°
- Horizontal launch from height h
- t = √(2h/g), x = v₀√(2h/g)
- Launched horizontally, θ = 0
Point particle launched and landing at the same height; uniform gravity and no air resistance.
Position, velocity & acceleration
- Velocity at time t
- v = v₀ + at
- The v–t graph has slope a
- Position at time t
- x = x₀ + v₀t + ½at²
- This lab starts at x₀ = 0
- Velocity without time
- v² = v₀² + 2a(x − x₀)
- Use when the time is unknown
- Average velocity
- v̄ = Δx/Δt = (v₀ + v)/2
- Second form needs constant a
- Displacement from a v–t graph
- Δx = area under the v–t graph
- Area below the axis counts as negative
- Turnaround time
- t = −v₀/a
- When v₀ and a have opposite signs
One-dimensional motion starting at x = 0 with constant acceleration.
Forces & energy formulas
Force & friction
- Normal force on a level surface
- N = mg
- No vertical acceleration
- Static friction
- fₛ ≤ μₛN
- Matches the push up to this limit
- Kinetic friction
- fₖ = μₖN
- Opposes the sliding motion
- Friction on level ground
- f = μmg
- Shortcut when N = mg
- Newton’s second law
- Fnet = ma
- Once sliding, a = (F − fₖ)/m
Horizontal surface, rightward applied force, g = 9.81 m/s². For comparison, static and kinetic coefficients both equal μ. The block starts at rest.
Energy on a ramp
- Gravitational potential energy
- PE = mgh
- h measured from a chosen reference level
- Kinetic energy
- KE = ½mv²
- Never negative
- Conservation of mechanical energy
- PE₀ + KE₀ = PE + KE
- No friction or drag
- Speed after a drop from rest
- v = √(2gΔh)
- Mass cancels
- Acceleration along the ramp
- a = g sin θ
- Sets the travel time, not the final speed
A sliding point mass starts from rest. No friction, air drag, or rotational energy; g = 9.81 m/s².
Buoyancy & floating
- Buoyant force (Archimedes’ principle)
- Fᵦ = ρfluid g Vsubmerged
- Weight of the displaced fluid
- Weight of the object
- W = ρobject g V
- Mass × g
- Floating fraction
- Vsubmerged / V = ρobject / ρfluid
- Only when ρobject < ρfluid
- Apparent weight underwater
- Wapparent = W − Fᵦ
- What a spring scale reads
Uniform fluid and object. Submerged fraction is set by the control. Only gravity and buoyancy are shown; drag and any external force holding the object are excluded. g = 9.81 m/s².
Waves formulas
Pendulum explorer
- Period (small angles)
- T = 2π√(L/g)
- Accurate below about 15°
- Frequency
- f = 1/T = (1/2π)√(g/L)
- Complete swings per second
- Length for a given period
- L = gT² / (4π²)
- Rearranged period formula
- Gravity from a measured period
- g = 4π²L / T²
- A classic lab measurement of g
- Angle over time
- θ(t) = θ₀ cos(2πt/T)
- Released from rest at θ₀
Small-angle approximation, massless rod/string, point bob, no damping. Angles are limited to 15°.
Spring oscillator
- Hooke’s law
- F = −kx
- Force points back toward equilibrium
- Period
- T = 2π√(m/k)
- Independent of amplitude
- Angular frequency
- ω = √(k/m)
- In rad/s; f = ω/2π
- Spring potential energy
- PE = ½kx²
- Largest at the turning points
- Total energy
- E = ½kA² = ½m·vmax²
- vmax = Aω at equilibrium
Horizontal ideal Hooke’s-law spring, no friction or damping. Displacement is measured from equilibrium.
Traveling wave explorer
- Wave speed
- v = fλ
- Frequency × wavelength
- Period
- T = 1/f
- Time for one complete cycle
- Wavelength
- λ = v/f
- Distance between neighbouring crests
- Traveling sine wave
- y = A sin(2π(x/λ − ft))
- Moves in the +x direction
An ideal transverse sinusoidal wave. Frequency and wavelength are independently set; wave speed changes with their product.
Electricity formulas
Ohm’s law & power
- Ohm’s law
- V = IR
- Voltage = current × resistance
- Current
- I = V/R
- Amperes from volts and ohms
- Resistance
- R = V/I
- Ohms from a measurement
- Power
- P = VI
- Watts = volts × amperes
- Power from current
- P = I²R
- Substitute V = IR
- Power from voltage
- P = V²/R
- Substitute I = V/R
Ideal DC voltage source and an ohmic resistor with constant resistance; wires have zero resistance.
Series & parallel resistors
- Series equivalent resistance
- R = R₁ + R₂ + …
- Same current through every resistor
- Parallel equivalent resistance
- 1/R = 1/R₁ + 1/R₂ + …
- Same voltage across every branch
- Two resistors in parallel
- R = R₁R₂ / (R₁ + R₂)
- “Product over sum”
- Series voltage divider
- V₁ = V · R₁ / (R₁ + R₂)
- Voltage drops add to the supply
- Parallel current divider
- I₁ = I · R₂ / (R₁ + R₂)
- Branch currents add to the total
Two ideal ohmic resistors, ideal DC source, zero-resistance wires.
Optics formulas
Refraction & total internal reflection
- Snell’s law
- n₁ sin θ₁ = n₂ sin θ₂
- Angles measured from the normal
- Refracted angle
- θ₂ = arcsin(n₁ sin θ₁ / n₂)
- No solution means total internal reflection
- Critical angle
- θc = arcsin(n₂/n₁)
- Only when n₁ > n₂
- Refractive index
- n = c/v
- Speed of light in vacuum ÷ speed in the medium
- Law of reflection
- θᵣ = θ₁
- Reflected angle equals incident angle
Flat interface between transparent media. Angles are measured from the normal; ray intensity and partial reflection are not modeled.
Converging lens explorer
- Thin lens equation
- 1/f = 1/dₒ + 1/dᵢ
- Real-is-positive sign convention
- Image distance
- dᵢ = f·dₒ / (dₒ − f)
- Negative dᵢ means a virtual image
- Magnification
- M = −dᵢ/dₒ = hᵢ/hₒ
- Negative M means inverted
- Lens power
- P = 1/f
- In dioptres when f is in metres
Ideal thin converging lens and paraxial rays. A positive object distance represents a real object; negative image distance represents a virtual image.
More essential physics formulas
Standard relationships that appear alongside the simulations in most introductory courses.
| Quantity | Formula | What the symbols mean |
|---|---|---|
| Newton’s second law | F = ma | Net force (N) = mass (kg) × acceleration (m/s²) |
| Weight | W = mg | g = 9.81 m/s² near Earth’s surface |
| Momentum | p = mv | Measured in kg·m/s |
| Impulse | FΔt = Δp | Force × time equals the change in momentum (N·s) |
| Work done by a force | W = Fd cos θ | θ is the angle between the force and the displacement |
| Power | P = W/t = Fv | Rate of doing work, in watts (W = J/s) |
| Centripetal acceleration | a = v²/r | Points toward the centre of the circle |
| Centripetal force | F = mv²/r | r is the radius of the circular path |
| Newton’s law of gravitation | F = Gm₁m₂/r² | G = 6.674 × 10⁻¹¹ N·m²/kg²; r is the distance between centres |
| Density | ρ = m/V | Measured in kg/m³ |
| Pressure | p = F/A | Measured in pascals (Pa = N/m²) |
| Fluid pressure at depth | p = ρgh | h is the depth below the surface |
| Electric charge | Q = It | Charge (C) = current (A) × time (s) |
| Efficiency | η = useful output ÷ total input × 100% | Works for energy or power |