DERIVATIONS
40 of the most important
derivations for A Level Physics
PhysicsOnline.com
FREE DOWNLOAD
A LEVEL
PHYSICS
Physics Online
Derivations | Introduction
Thank you for downloading this guide, I really hope you find it useful!
I have brief derivations for forty of the most important equations that you may come
across during your A Level Physics course. Some of these are listed in the specification,
many have come up in previous exam questions and the others just help you understand
the underlying physics.
Derivation | Period of a Pendulum
Don’t forget that I also have short videos where I explain each of these derivations in a
little more detail at ALevelPhysicsOnline.com. There are also worksheets (with varying
levels of support) you can use when you’re revising these.
1
2
While you’re on the website, don’t forget that you can sign up for a Premium Plan (or
your teachers can buy a School Subscription) to access hundreds more videos and
resources.
There is also a workbook called the Daily Workout with short questions you can complete
every single day – available to by on Amazon.
AL e v e l P h ys i c s O nl i ne .c om / d e r i v a ti o n s
Derivations | Contents
▪ Electrical Power
▪ Capacitors in Series
▪ Resistors in Series
▪ Capacitors in Parallel
▪ Resistors in Parallel
▪ Closest Approach
▪ Two Resistors in Parallel
▪ Coulombs Law
▪ Drift Velocity
▪ Gravitational Attraction
▪ Gravitational Potential Energy
▪ Gravitational Field Strength
▪ Kinetic Energy
▪ Radial Path of a Particle
▪ Elastic Potential Energy
▪ Velocity Selector
▪ Kinetic Energy and Momentum
▪ Centripetal Acceleration
▪ Power, Force and Velocity
▪ Orbital Period
▪ suvat Equations
▪ SHM
▪ Newton's 2nd Law
▪ Total Energy of a Satellite
▪ Upthrust
▪ Escape Velocity
▪ Springs in Series
▪ Period of a Pendulum
▪ Springs in Parallel
▪ Period of a Spring-Mass
▪ Diffraction Grating
▪ Ideal Gas Equation
▪ Double Slit
▪ Ek and Temperature
▪ Double Slit (for AQA)
▪ Kinetic theory
▪ Half-life
▪ The Parsec
▪ Capacitor Energy
▪ Specific Charge
AL e v e l P h ys i c s O nl i ne .c om / d e r i v a ti o n s
Derivation | Electrical Power
Derive the series of expressions linking power, current, potential
difference and resistance:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
P = I2 R
P = V2/ R
Derivation | Resistors in Series
Derive the expression for calculating the total resistance
of resistors in series:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
RT = R1 + R2 + … + Rn
Derivation | Resistors in Parallel
Derive the expression for calculating the total resistance
of resistors in parallel:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
1 ___
1
1
1
___
=
+ ___ + … + ___
RT
R1
R2
Rn
Derivation | Two Resistors in Parallel
Derive the expression for calculating the total resistance
of resistors in parallel:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
R1 R2
RT = ________
R1 + R2
Derivation | Drift Velocity
Derive the expression for calculating the drift velocity of charge carriers:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
I = Ane v
Derivation | Potential Energy
Derive the expression for change in gravitational potential
energy in a uniform field:
ALevelPhysicsOnline.com/derivations
Ep = m g h
Derivation | Kinetic Energy
Derive the expression for kinetic energy:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Ek = ½ m v2
Derivation | Elastic Potential Energy
Derive the expression for elastic potential energy stored in a spring:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Ee = ½ k x2
Derivation | Energy and Momentum
Derive the expression for kinetic energy in terms of momentum and mass:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
p2
Ek = ____
2m
Derivation | Power, Force and Velocity
Derive the expression for power in terms of force and velocity:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
P = Fv
Derivation | ‘suvat’ Equations
Derive the ‘suvat’ equations for an object
undergoing uniform acceleration:
v = u + at
s = u t + ½a t2
s = ½ (u + v) t
v2 = u2 + 2 a s
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Newton’s 2nd Law
Derive the expression to show how the resultant force is related
to the rate of change of momentum:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Δp
F = ____
Δt
Derivation | Upthrust
Derive the expression for the upthrust on an object submerged in a fluid:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
U = ρVg
Derivation | Springs in Series
Derive the expression for calculating the combined
stiffness of springs in series:
1 ___
1
1
1
___
=
+ ___ + … + ___
kT
k1
k2
kn
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Springs in Parallel
Derive the expression for calculating the combined
stiffness of springs in parallel:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
kT = k1 + k2 + … + kn
Derivation | Diffraction Grating
Derive the expression for a diffraction grating:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
d sinθ = n λ
Derivation | Double Slit
Derive the expression for the fringe spacing from a double slit:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
ax
λ = ___
D
Derivation | Double Slit (AQA)
Derive the expression for the fringe spacing from a double slit:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
λD
w = ___
s
Derivation | Half-life
Derive the expression for the half-life of a radioactive isotope
in terms of its decay constant:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
ln2
t½ = ____
λ
Derivation | Energy in a Capacitor
Derive the series of expressions for the energy stored by a capacitor
to its capacitance, charge stored and potential difference:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
E = ½CV2
E = ½Q2/ C
Derivation | Capacitors in Series
Derive the expression for calculating the total capacitance
of capacitors in series:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
1 ___
1
1
1
___
=
+ ___ + … + ___
CT C1 C2
Cn
Derivation | Capacitors in Parallel
Derive the expression for calculating the total capacitance
of capacitors in parallel:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
CT = C1 + C2 + … + Cn
Derivation | Closest Approach
Derive the expression for the distance of closest approach
for an alpha particle to a nucleus:
ALevelPhysicsOnline.com/derivations
qQ
r = _________
2 π ε0 m v2
Derivation | Coulomb’s Law
Derive Coulomb’s law between two charged objects:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
qQ
F = _______
4 π ε0 r2
Derivation | Gravitational Attraction
Derive Newton’s law of universal gravitation between two objects:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
- GM m
F = ________
r2
Derivation | Gravitational Field Strength
Derive the expression for gravitational field strength at a
distance from a massive body:
-GM
g = ______
r2
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Radial Path
Derive the expression for the radius of a path of a
charged particle in a magnetic field:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
p
r = ____
BQ
Derivation | Velocity Selector
Derive the expression for the velocity of a particle travelling in a
straight line in a magnetic and electric field:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
E
v = ___
B
Derivation | Centripetal Acceleration
Derive the expression for centripetal acceleration:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
v2
a = ____
r
Derivation | Orbital Period and Radius
Derive the expression for the relationship between orbital
period squared and orbital radius cubed:
4 π2 r3
______
2
T =
ALevelPhysicsOnline.com/derivations
GM
Derivation | Simple Harmonic Motion
Show that for a system undergoing simple harmonic motion:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
a ∝ -x
Derivation | Total Energy of a Satellite
Derive the expression for the total energy of a satellite:
- Gm M
ET = ________
2R
AL e ve l Ph y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Escape Velocity
Derive the expression for the escape velocity from a planet:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
v=
2GM
______
r
Derivation | Period of a Pendulum
Derive the expression for time period for a pendulum undergoing SHM:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
l
T = 2 π ___
g
Derivation | Period of a Spring-Mass
Derive the expression for period for a spring-mass system undergoing SHM:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
m
T = 2 π ___
k
Derivation | Ideal Gas Equation
Derive the ideal gas equation:
pV = nRT
pV = NkT
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Ek and Temperature
Derive the expression relating the kinetic energy of a particle
to the temperature of an ideal gas:
ALevelPhysicsOnline.com/derivations
3
Ek = __ k T
2
Derivation | Kinetic Theory
Derive the kinetic theory formula for an ideal gas:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
1
_
p V = __ N mc2
3
Derivation | The Parsec
Derive the parsec and show that:
1 pc = 3.1 x 1016 m
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
Derivation | Specific Charge on an eDerive the expression for the specific charge on an electron:
AL e ve l P h y s i c s O n l i n e .c o m / d e r i v a t i o n s
e
2V
__
= ____
m B2 r2