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SAT Formula Reference

This guide is a compilation of about ﬁfty of the importantphysicsformulastoknow for the SAT Subject in physics. (Note that formulas are not given on the test.) Each formula row contains a description of the variables or constants that up the formula, along with a brief explanation of the formula.

Kinematics

vave =averagevelocity ∆x The deﬁnition of average ve- vave = ∆x = ∆t locity. ∆t =elapsedtime

vave =averagevelocity (v + v ) Another deﬁnition of the av- v = i f ave 2 =initialvelocity erage , which works a vf =ﬁnalvelocity when is constant.

a = ∆v a = ∆t ∆v =changeinvelocity The deﬁnition of acceleration. ∆t =elapsedtime

∆x =displacement v 1 2 i =initialvelocity Use this formula when you ∆x = vi∆t + a(∆t) 2 ∆t =elapsedtime don’t have vf . a =acceleration

∆x =displacement

vf =ﬁnalvelocity 1 2 Use this formula when you ∆x = vf ∆t − a(∆t) 2 ∆t =elapsedtime don’t have vi. a =acceleration

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Kinematics (continued)

vf =ﬁnalvelocity

2 2 vi =initialvelocity Use this formula when you vf = vi +2a∆x a =acceleration don’t have ∆t. ∆x =displacement

Dynamics

F = ’s Law. Here, F = ma m = F is the on the mass a =acceleration m.

W = The weight of an with m W = mg =mass mass m.Thisisreallyjust g =accelerationdue Newton’s Second Law again. to

f =frictionforce The “Physics is Fun” equa- tion. Here, µ can be either µ =coeﬃcient f = µN the kinetic coeﬃcient of fric- of tion µ or the static coeﬃcient N =normalforce k of friction µs.

p = The deﬁnition of momentum. p = m =mass It is conserved (constant) if there are no external on v =velocity asystem.

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Dynamics (continued)

∆p =change in momentum ∆p = F ∆t F ∆t is called the . F =appliedforce ∆t =elapsedtime

Work, , and

W = F =force W = Fdcos θ Work is done when a force d = is applied to an object as it or moves a distance d. F is the θ =anglebetweenF ! component of F in the direc- W = F!d and the direction tion that the object is moved. of

F! =parallelforce

KE = 1 The deﬁnition of kinetic en- KE = mv2 2 m =mass ergy for a mass m with veloc- v =velocity ity v.

PE = energy m =mass The for a PE = mgh g =accelerationdue mass m a height h above to gravity some reference level. h =height

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Work, Energy, Power (continued)

The “work-energy” theorem: W W =∆(KE) =workdone the work done by the net force KE = kinetic energy on an object equals the change in kinetic energy of the object.

E=totalenergy The deﬁnition of total (“me- chanical”) energy. If there E=KE+PE KE = kinetic energy is no friction, it is conserved PE = potential energy (stays constant).

P =power Power is the amount of work W P = W =work done per unit (i.e., power ∆t is the at which work is ∆t =elapsedtime done).

Circular Motion

The “centripetal” acceleration 2 ac =centripetalacceleration v for an object moving around ac = v =velocity r in a of r at veloc- r =radius ity v.

Fc =centripetalforce The “centripetal” force that is 2 mv m =mass needed to keep an object of Fc = r v =velocity mass m moving around in a circle of radius r at velocity v. r =radius

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Circular Motion (continued)

v =velocity This formula gives the veloc- 2πr v = r =radius ity v of an object moving once T around a circle of radius r in T =period time T (the period).

1 f = The frequency is the number f = of per second that an T T =period object moves around a circle.

Torques and

τ = τ = rF sin θ r =distance(radius) Torque is a force applied at a distance r from the axis of ro- or F =force tation. F⊥ = F sin θ is the θ F τ = rF⊥ =anglebetween component of F perpendicu- and the arm lar to the lever arm. F⊥ =perpendicularforce

L =angularmomentum Angular momentum is con- m =mass served (i.e., it stays constant) L = mvr v =velocity as long as there are no exter- nal . r =radius

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Springs

Fs =springforce “Hooke’s Law”. The force is k =springconstant Fs = kx opposite to the stretch or com- x =springstretchor pression direction.

The potential energy stored PEs =potentialenergy in a when it is ei- k =springconstant 1 2 ther stretched or compressed. PEs = kx 2 x =amountof Here, x =0correspondsto spring stretch the “natural ” of the or compression spring.

Gravity

Fg =forceofgravity Newton’s Law of Gravitation: G =aconstant m1m2 this formula gives the attrac- Fg = G 2 r m1,m2 = tive force between two masses r =distanceof adistancer apart. separation

Electric Fields and Forces

Fe =electricforce “Coulomb’s Law”. This for- k =aconstant q1q2 mula gives the force of attrac- Fe = k 2 r q1,q2 =charges tion or repulsion between two r =distanceof charges a distance r apart. separation

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Electric Fields and Forces (continued)

Achargeq,whenplacedinan F =electricforce electric ﬁeld E,willfeelaforce F = qE E =electricﬁeld on it, given by this formula (q is sometimes called a “test” q = charge, since it tests the elec- tric ﬁeld strength).

E =electricﬁeld This formula gives the elec- k =aconstant tric ﬁeld due to a charge q at q adistancer from the charge. E = k 2 q =charge r Unlike the “test” charge, the r =distanceof charge q here is actually gen- separation erating the electric ﬁeld.

Between two large plates of metal separated by a distance E =electricﬁeld V d which are connected to a E = V = d battery of voltage V ,auni- d =distance form electric ﬁeld between the plates is up, as given by this formula.

The potential diﬀerence ∆V ∆V =potentialdiﬀerence between two points (say, the W terminals of a battery), is de- ∆V = W =work q ﬁned as the work per unit q =charge charge needed to move charge q from one point to the other.

Circuits

’s Law”. This law gives V =voltage the relationship between the V = IR I = battery voltage V ,thecurrent R =resistance I,andtheresistanceR in a circuit.

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Circuits (continued)

P = IV P =power All of these power formulas or are equivalent and give the 2 I =current P = V /R power used in a circuit V =voltage or R.Usetheformulathathas R =resistance the that you know. P = I2R

Rs =total(series) When are placed end resistance to end, which is called “in se- Rs = R1 =ﬁrstresistor ries”, the eﬀective total resis-

R1 + R2 + ... R2 =secondresistor tance is just the sum of the in- ... dividual resistances.

When resistors are placed side 1 Rp =total(parallel) = resistance by side (or “in parallel”), the Rp eﬀective total resistance is the R1 =ﬁrstresistor inverse of the sum of the re- 1 1 R2 =secondresistor + + ... ciprocals of the individual re- R1 R2 ... sistances (whew!).

This formula is “Ohm’s Law” for capacitors. Here, C is a q =charge number speciﬁc to the capac- q = CV C =capacitance itor (like R for resistors), q is V =voltage the charge on one side of the capacitor, and V is the - across the capacitor.

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Magnetic Fields and Forces

This formula gives the force F =forceonawire on a wire carrying current I I =currentinthewire while immersed in a magnetic L =lengthofwire ﬁeld B.Here,θ is the F = ILB sin θ between the direction of the B =externalmagneticﬁeld current and the direction of θ =anglebetweenthe the magnetic ﬁeld (θ is usu- current direction and ally 90◦,sothattheforceis the magnetic ﬁeld F = ILB).

The force on a charge q as it F =forceonacharge travels with velocity v through q =charge amagneticﬁeldB is given by v =velocityofthecharge this formula. Here, θ is the F = qvB sin θ angle between the direction of B =externalmagneticﬁeld the charge’s velocity and the θ =anglebetweenthe direction of the magnetic ﬁeld direction of motion and (θ is usually 90◦,sothatthe the magnetic ﬁeld force is F = qvB).

Waves and

This formula relates the - v =wavevelocity length and the frequency of a v = λf λ =wavelength wave to its . The for- f =frequency mula works for both and .

When light travels through a v =velocityoflight medium (say, glass), it slows c down. This formula gives the v = c =vacuumlightspeed n in a medium n =indexofrefraction that has an index of refraction n.Here,c =3.0 × 108 m/s.

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Waves and Optics (continued)

“Snell’s Law”. When light moves from one medium (say, n1 =incidentindex air) to another (say, glass) θ1 =incidentangle with a diﬀerent index of re- n1 sin θ1 = n2 sin θ2 n2 =refractedindex n,itchangesdirec- tion (refracts). The are θ2 =refractedangle taken from the (per- pendicular).

This formula works for lenses 1 1 1 do =objectdistance + = and mirrors, and relates the di =imagedistance do di f focal length, object distance, f =focallength and image distance.

The magniﬁcation m is how much bigger (|m| > 1) or d m =magniﬁcation m = − i smaller (|m| < 1) the image di =imagedistance do is compared to the object. If do =objectdistance m<0, the image is inverted compared to the object.

Heat and

The speciﬁc c for a sub- Q =heatadded stance gives the heat needed or removed to raise the of a m =massofsubstance Q = mc ∆T mass m of that substance by c =speciﬁcheat ∆T degrees. If ∆T<0, the formula gives the heat that ∆T =changein has to be removed to lower the temperature temperature.

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Heat and Thermodynamics (continued)

When a substance undergoes achangeofphase(forexam- Q =heatadded ple, when ice melts), the tem- or removed perature doesn’t change; how- Q = ml m =massofsubstance ever, heat has to be added (ice l =speciﬁcheat melting) or removed (water of transformation freezing). The speciﬁc heat of transformation l is diﬀerent for each substance.

∆U =changein The “ﬁrst law of thermody- namics”. The change in inter- − ∆U = Q W Q =heatadded nal energy of a system is the W =workdone heat added minus the work by the system done by the system.

Aheatengineessentiallycon- Eeng =%eﬃciencyof verts heat into work. The the heat engine does work by absorb- W ing heat from a hot reservoir Eeng = × 100 W =workdone Qhot by the engine and discarding some heat to acoldreservoir.Theformula Qhot =heatabsorbed by the engine gives the (“eﬃciency”) of the engine.

Pressure and Gases

P = The deﬁnition of pressure. P F P = F =force is a force per unit exerted A by a gas or ﬂuid on the walls A =area of the container.

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Pressure and Gases (continued)

The “ Law”. For “ideal” gases (and also for P =pressure PV real- gases at low pressure), =constant V = T the pressure of the gas times T =temperature the volume of the gas divided by the temperature of the gas is a constant.

Modern Physics and Relativity

E =photonenergy The energy of a is proportional to its wave fre- E = hf h =aconstant quency; h is a number called f =wavefrequency “Planck’s constant”.

Aparticlecanactlikeawave h λ =matterwavelength with wavelength λ,asgivenby λ = p h =aconstant this formula, if it has momen- p =momentum tum p.Thisiscalled“wave- ” duality.

The relativistic factor γ is the amount by which moving γ =therelativisticfactor 1 slow down and γ = v =speedofmoving !1 − (v/c)2 contract, as seen by an ob- observer server compared to those of c =speedoflight another observer moving at speed v (note that γ ≥ 1).

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