A.P. Physics (B) Exam “Quick Reference” Study Guide

This study guide is to provide you with a source of all the information that I [Mr. Welkley] consider the MOST important for you to have essentially memorized for Part I of the A.P. Exam, whether it is formulas or concepts. This stuff you have to know cold! If you do, you will save a great amount of in answering questions without having to go through a lot of calculating.

Linear Memorize the following formulas:  When an object is accelerating there is a SQUARED relation between traveled ∆x = v t + 1 at 2 o 2 and the time spent traveling: If the time an accelerating object travels is tripled, the 2 2 v = vo + 2a∆x distance it travels while accelerating will be nine time further than original.  It is a DIRECT relation between and time when there is a constant . v = vo + at  The slope of a distance/time graph is the average velocity.  The slope of a velocity/time graph is the average acceleration, while the under the graph is the .

SPECIAL CASES of Linear Motion

FREE FALL – Substitute the acceleration due to (g) into each of the three linear motion equations. Also remember to use 10 m/s 2 for the value of g in those equations to make quick calculations.  Time to fall from rest: rework The first linear motion equation to solve for the time to fall from a certain 2y height and it becomes: t = where y is the height fallen from. g  or Velocity as the object strikes the ground from a known height: rework the linear motion equation to solve for speed as the object strikes the ground and you get: v = 2gh where h is the height the object falls from.

PROJECTILE MOTION: You need to remember the two situations: Starting by being thrown horizontally ( off a table edge), and being kicked off the ground at an (propelled up at an angle.) Horizontal Projection:  Remember that the time to fall (from X to Y) is the same as vx 2y X the time of flight . Use: t = . This determines how long g the object is in the air. Y Z  Remember that something falling from X to Y takes the same time to move as the object flying from X to Z.

 To find the distance from Y to Z using: x = vxt where t is the time to fall.  Increasing vx Does NOT increase the time of flight, only increasing the height does! And, that is a square root relation. To double the flight time, the height must be increased by four . If the height is what is doubled, the flight time will only increase by 2 . Y P Regular Projectile Motion:  The acceleration is always g and is always downward throughout the flight. X θ Z  The velocity at Y is NOT zero, but is the horizontal component of the object’s launch velocity.  Determine the time of flight using the vertical components of motion. Hangtime is the total time it’s in the air and it is two times the time it takes to rise to point Y.

 To find the RANGE of the flight: x = vxt  As the angle of launch increases for a given initial launch speed: 1. The horizontal component of the launch velocity decreases. 2. The vertical component of the launch velocity increases. 3. The height the object rises to increases. 4. The time of flight increases. 5. The RANGE (horizontal distance (x)) traveled increases until the launch angle is 45º then it decreases 6. Complimentary of launch will give you the same RANGE for the projectile to travel.

VECTORS and TRIG Functions

30-60-90 Right Triangle: let’s say two vectors are acting perpendicularly to one another and the angle between one of them and the resultant is either 30º or 60º. They will automatically form a 30-60-90 Rt. Triangle. When that happens there is the following relation between all the sides and parts of the triangle and the vectors. Angle Sine Cosine Tangent 1. Vector B must be 3 times the length of Vector 1 3 1 3 Resultant A. 30 or 2. The Resultant will always be 2 times the length 2 2 3 3 Vector BVector of Vector A. 3 1 3. Memorize the following Trig relations for this 60 3 2 2 60º triangle especially for SINE : Vector A

Golden Triangle: 3-4-5 Rt. Triangle: Let’s say that two vectors are acting perpendicularly to each other and they have a relation between their lengths where the ratio of the short to the long leg of the triangle is 3:4 (i.e. 12:16, or 21:28, or 1.5:2). The triangle formed by these two 37º vectors and their resultant make the 3-4-5 Golden Triangle and have the following relation: Resultant = “5” 1. The angle opposite the “3” side will always be Angle Sine Cosine Tangent 37º. 3 4 3 2. The angle opposite the “4” side will always be 37 Vector B Vector = “4” 53º. 5 5 4 53º 4 3 4 3. Memorize the following trig relations for this 53 Vector A = “3” triangle especially for SINE : 5 5 3

Newton’s Laws: 1. Be able to draw the Free-Body Diagram of all the acting on an object. 2. USE the Free-Body Diagram to find the Net acting on an object. 3. Forces that act Parallel to each other can be added or subtracted to find the . Only forces parallel

to the direction of motion will create a linear acceleration that is changing the speed. Fnet = ∑ F = mTotal a 4. Forces that act Perpendicular to the motion can change the direction of the motion and may create curved motion. Perpendicular forces create and centripetal or “radial” acceleration. 5. always acts opposite to the motion or potential motion of the object. 6. The Normal Force is the force of the surface pressing up on the object perpendicular to the surface, not necessarily the ground. 7. Tension is a force!

Circular Motion:

2 1. has a squared relation with the speed the object is moving around the circle mv with. FC = r 2. Centripetal Force has an inverse relation with the object’s radius. v2 3. Centripetal Force is ALWAYS the NET force acting on the rotating object. a = c r 4. Centripetal acceleration is also called radial acceleration. F = ma 5. BOTH centripetal force and centripetal or radial acceleration ALWAYS act toward the center C c of the circle or curve made by the object. 6. The direction of the instantaneous velocity is always perpendicular to the centripetal force and centripetal/radial acceleration. 7. If the speed is NOT constant, then the object is experiencing TANGENTIAL (or linear) acceleration in addition to the radial acceleration. The Total acceleration that the object experiences is the vector sum of these two . Since they are perpendicular to each other (tangential is ⊥ to centripetal/radial)

2 2 at this means that: aTotal = ac + at and the angle between them is: tanθ = ac 8. Horizontal circle on a rough table top: Set the centripetal force equal to the frictional force (because friction is making it turn!) and the minimum speed for a given centripetal force is v = µgr

9. VERTICAL Circle: Remember that the centripetal force must be F net .

A) At top: FC = FA + Fg . At a certain minimum speed F A becomes zero (there is no tension on the rope holding the pail, the passengers in the roller coast “just remain” in their seats, the block “remains in contact” with the track at the top of the loop, etc.) then FA at the top is ZERO and solving for the minimum speed we get: v = gr

B) At the Bottom: FC = FA − Fg . F A is the “Tension on the rope, the force the riders in the roller coaster

feel, etc.” at the bottom of the circle. Thus: FA = FC + Fg this is showing why you feel heavier at the bottom of the track.

Gravitation: The force of gravity is an inverse square relation with the distance between the centers of the objects. When dealing with earth, they will often talk about in terms of a number of “Earth Radii” above the earth’s surface. Don’t forget we already stand ONE EARTH RADIUS from the earth’s center. T 2 Planetary Motion: = k . Orbital Period (T) Mean orbital radius (r). This is a constant for any set of r3 objects orbiting another one.

Work--Conservation of Energy-Linear Motion 1. Remember that for to be done, there must be a change in energy in the system. W = F ⋅ d cosθ Going around a circle “does no work” if it is done at a constant speed. W P = = F ⋅v 2. When a force acts to “do work” if it is not parallel to the motion (displacement) you need t to use the cosine of the angle between the force and displacement vectors (this gives you 1 2 the component of the force acting parallel to the displacement!) KE = 2 mv 3. KE and speed/velocity have a squared relation. 1 2 U Spring = 2 kx A) Remember, if Velocity is changed…square that change to find the change in KE. B) If the KE is changed…take the square ROOT of the change in KE to find the change UGravity = mgh in v. 4. When there is no Friction, the TOTAL energy must remain constant, thus KE + any PE will be the same at the start and at the finish. This means if KE decreases, PE must increase by the same amount. 5. If there IS friction then W = Ff ⋅d = ∆KE Friction is always parallel to motion, but decreases the overall energy of the system!

Momentum: p = mv 1. always has the same direction as velocity. ∆p = m∆v 2. (F·t) must cause a change in momentum. 3. The impulse that one object exerts on another in a collision is the same regardless of F ⋅t = ∆p which object is doing the colliding or the of each object (N’s Third Law!) p + p = p + p 1 2 1 f 2 f 4. Inelastic Collision when one is motionless: p1 = (m1 + m2 )v f and if both are the same, they both move off at half the original speed of the moving object. 5. RECOIL: A) The total momentum is Zero before and after the gun/bullet, cannon/shell, boy/girl move away from each other.

B) p1 = − p2 6. Elastic Collision: Both Momentum AND Energy must be conserved.

Static Electricity: 1. Electrostatic Force has an inverse square relation with distance between two charged kQ Q 1 2 objects. FE = 2 d 2. The direction of the electric field is defined as the direction a positive “test” charge would kQ F E = = E move in the field with respect to the charge creating the field. Generally this is From d 2 Q Positive to Negative. V 3. Electric Potential (Voltage) is the amount of energy (W) needed to move a charge (Q) E = through an electric Field. d 4. The Electric Potential of a field increases as you move toward the (+) side of the field. W V = High Potential is at the highest (+) location in the field. Q 5. Equipotential Lines are always perpendicular to the Electric Field lines. Equipotential lines are lines of “EQUAL” voltage in an electric field. Moving a charge along those lines does not change its energy.

Electric Circuits: V V  Ohm’s Law works for finding voltage, resistance, or current through an R = or V = IR or I = individual component of the circuit, or for the source voltage, total current, I R or effective resistance as a whole. RSeries = ∑ R  Resistance depends not only on current and voltage, but it can be found 1 1 knowing a conductor’s length, composition, cross sectional area, and = temperature. R ∑ R Parallel  Kirchoff’s Voltage Law (Loop Theorem): Source Voltage equals the sum of P = VI = I 2 R all voltage drops in a loop of the circuit. Explains what happens in a Series Q Circuit. C = V  Kirchoff’s Current Law (Junction Theorem): Sum of current entering a junction must equal the sum of currents leaving the junction in a circuit. C = C Parallel ∑ Explains the Total current and Branch current properties of a Parallel 1 1 Circuit. = ∑  Series Circuit: CSeries C o Effective resistance equals the sum of all the resistances in the circuit. 1 2 U = 2 CV o Each component in the circuit experiences the same current through them. BUT if you change the circuit, the current in the circuit will change. Each component will then experience this new current. o As loads are added effective resistance increases and total current decreases.  Parallel Circuit: o Effective resistance is the “Sum of the Reciprocals” of the branch resistances. o Each branch experiences the same voltage (usually it is the same as the source voltage if they are all wired in parallel to the source.) o Total current is the sum of all the branch currents. o As loads are added in parallel the total current increases while the effective resistance decreases.  Capacitors store electric charge (Q) based on the voltage across them (V).  Capacitors wired in series act like resistors in parallel when determining total capacitance of a circuit.  Capacitor wired in parallel act like resistors wired in series when determining total capacitance.

Magnetism: 1. Magnetic Field around a current carrying wire: Thumb is “I” and fingers curl in direction F = BIL of the magnetic field. F = Bqv 2. Magnetic Field in a loop of wire: Fingers curl in direction of current (I) in the wire, Thumb ε = BLv points the direction of the magnetic field (B) INSIDE or in the center of the loop. ∆φ ∆BA 3. Force acting on a current Carrying wire in a magnetic field uses Flat Right Hand ε = = (Electric Motor) ∆t ∆t A) Palm is the Force (F) acting on the wire ε V = B) Fingers are the magnetic field direction (B) eff 2 C) Thumb is the current flowing in the wire (I) 4. Electromagnetic Induction (Electric Generator) Current being induced in a wire moving ⊥ to the magnetic Field, or the magnetic field moving perpendicularly to the plane of the are of the loop of wire: Left Hand Rule A) Palm is the direction of Motion/Force in the wire/Direction of magnetic field change (magnetic flux). B) Fingers are the direction of the magnetic Field C) Thumb is the direction of the induced Current. 5. Effective Voltage or Current of an A.C. generator is 71% ( 1 ) of the max EMF (V) or current generated 2 by the device.

Fluid  Pressure is Force over an Area. 2 P = P0 + ρ gh 1  Bernoulli’s Principle [P + ρgy + 2 ρv = const.] means that the pressure Fbuoy = ρ Vg (P) on the fluid, the pressure factor of the height (ρgy ) and the pressure 2 1 2 1 factor of its motion ( 2 ρv ) are the same through out the fluid. If you P + ρ gy + 2 ρv = const. increase: 1. The Pressure – Either the fluid height will increase ( y) or the fluid will move faster ( v). 2. The Height/Depth – Either the pressure ( P) will increase at the bottom, or the fluid will flow faster at the bottom. 3. Speed of flow – Either the pressure of the fluid in with the flow will increase, or the fluid will shoot to a higher height.  Buoyancy ( Fbuoy ) depends on the volume of water displaced by the object. If the density of the object is less than the water (or liquid) then it will float and the ratio of the density of the object to the density of the liquid determines how deep the object will sink.

Thermodynamics  Try to remember the Universal Gas Law [ PV = nRT ] in terms of the “Named” Gas Laws  Charles Law: Constant Pressure – Temperature and Volume have a direct relation. PV = nRT = Nk T B This is also called an Isobaric change. W = −P∆V  Boyles Law: Constant Temp – Pressure and Volume are inversely related. This is ∆U = Q +W called an Isothermic change.  Gay-Lussac’s Law: Constant Volume – Pressure and Temperature are directly W e = related. This is called an Isochonic change. QH  See the “Graphing Section” regarding Pressure/Volume graphs.  The thermal energy of a system (U) depends on the Heat added (Q) and the Work of TH −TC eC = the system. If the work is done by the gas, then W is negative. If the work is done on T H the gas, then W is positive.  The efficiency of a system is the ration of the work done to the Heat gained by the system.  The Carnot efficiency is based on the difference between temperature of the gas when heated and cooled.

Simple Harmonic Motion:  Period of of a mass on a spring is directly proportional to the square root 1 m T = TS = 2π of the mass on the spring and inversely proportional to the square root of the f k spring constant. 2π L  Period of oscillation of a is directly proportional to the square root of ω = T = 2π T P g the length of the pendulum and inversely proportional to the local acceleration due to gravity. x = Acos()ωt  Vibrating objects experience their greatest acceleration at the maximum amplitude  2π  of vibration. This is also the of maximum . vmax =  A  Vibrating objects experience their maximum velocity (and therefore maximum  T  ) when it is moving through the equilibrium position. 2  2π  amax =   A  T 

E = 1 kA2 2

Waves and Light: λ 1. All waves behave the same way with respect to reflection, refraction and diffraction. v = fλ = 2. For Light, the index of refraction indicates how much a medium slows the speed of light down. T 3. For Sound: Resonance: Closed Pipe L= ¼ λ Open Pipe L= ½ λ and each odd multiple of v n = these wavelengths. c 4. For Sound: Doppler Effect: Moving toward you pitch/ is higher. Moving away pitch is lower. 5. For Light: Doppler Effect: Moving toward shifted toward “Blue” wavelengths. Moving away shifted toward “Red” wave lengths. 6. Constructive Interference: Amplitudes add – Sound gets louder, light gets brighter. Always happens when waves are “In Phase” Minimum distance for constructive interference is every full wavelength. 7. Destructive Interference: Amplitudes subtract – Sound gets softer, light gets dimmer/darker. Happens with waves are “180º out of phase”. This happens every odd half wavelength (i.e. λ , 3λ , 5λ , etc.) 2 2 2 8. Standing Wave: Nodes=Destructive Interference, Antinodes=Constructive Interference 9. Refraction: A) When the wave slows down as it enters the new medium: θi>θr (going from low index to high index, or “less dense” to “more dense”.) Frequency remains the same . B) Know how to determine the comparative index or “density”/Permittivity of mediums based on the path of the wave as it is refracted moving from one medium to another. C) Index of refraction for light is the ratio of the speed of light in a vacuum to the speed of light in a medium. 10. Critical Angle: when angle of incidence causes an angle of refraction of 90º. If angle of incidence is > critical angle: Total Internal Reflection. This only happens when going from a high index medium to a low index medium. 11. Optics: A) Know the four cases of how a concave mirror and a thin convex lens produce images based on the position of the object compared to the optical devices focus point and double focus distance (C and 2F). B) Know when a virtual and real image are produced and where they are located.

Modern Physics: 1. Photoelectric Effect: E = hf A) A Photon is “essentially” a particle of light energy. h p = B) The energy of the photon depends on the frequency of the photon. λ C) The momentum of the photon depends on its wavelength. E = mc2 D) Photoelectric materials “Turn On” (release an electron) when a certain amount of energy (threshold frequency) hits them. This energy is called the work function . E) If the photon has a frequency higher than the threshold frequency, the energy of the electron (its voltage) is greater than zero. F) If the number of photons (brightness of the light) increases the number of electrons emitted increases (current in the photocell is higher.) 2. Photon momentum: photons do not have mass, but can affect the momentum of massive particles (i.e. electrons). The change in momentum of the particle corresponds to the increase in wavelength of the photon. 3. De Broglie Wavelength – Matter Waves: Only very small or very slow particles can exhibit a wavelength. The wavelength of a photon is based on its momentum. 4. Emission Spectra/Absorption Spectra: The colored lines emitted by an excited atom are the of the photon emitted by the atom as electrons transition from higher energy levels to lower energy levels. The frequency of these photons corresponds to the energy of the photon and is equal to the difference in the

energy of each energy level in the atom. E Photon = Ei − E f

Nuclear Physics: 1. Atomic Number (A) = Number of protons m i 2. Mass Number (Z) = Number of protons + number of neutrons m f = 2 n 3. Halflife is the amount of time for half of the atoms of an isotope of an element n = Number of halflives to decay into a new isotope of a different element. 4. Radioactivity is proportional the number of atoms present decaying. 5. Alpha Decay – the alpha particle is a Helium nucleus, A decreases by 2, Z decreases by 4. 6. Beta Decay – the beta particle is a high energy electron released when a neutron decays into a proton and the beta particle. A increases by 1, Z remains the same. 7. Energy from mass: The change in the mass of a particle corresponds to the change in energy of a system by E = mc 2

Graphs: This is an incredibly important part of all aspects of the test and topics we have covered. Linear Motion Graphs: 1. Distance vs. Time graphs A) Constant Velocity – Linear – Slope = Velocity B) Constant Acceleration – Squared or Parabolic – Slope is instantaneous velocity (Tangent to curve.) C) No motion – Horizontal Line 2. Velocity vs. Time graphs A) Slope = acceleration B) Area under graph (between graph and x-axis) = displacement 3. Acceleration vs. Time graphs A) Area = Average change in Velocity B) Slope is rate of change of acceleration.

Force Graphs: 1. Force vs. Time graphs – indicate the Impulse acting to change momentum/motion. A) Area = Impulse i. Impulse = change in momentum ii. Impulse divided by mass = change in velocity B) Slope of this graph is meaningless for our purposes. It is an indication of how much the force is changing over time. 2. Force vs. Distance/Displacement graphs A) Area = Work done B) Special Case: Spring graphs – Force vs. elongation/stretch distance. i. Slope = spring constant ii. Area = work done stretching spring = change in potential energy stored in the spring

Ohm’s Law: Usually a Voltage (y-axis) vs. Current (x-axis) graph. A) Slope = Resistance B) Area =

Pressure/Volume: A) A vertical line on the PV graph indicates an Isochonic change (No change in volume), thus the change in pressure corresponds to a change in temperature. Heat is being added or taken away to do this. B) A horizontal line on the PV graph indicates an Isobaric change (No change in pressure), thus the change in volume corresponds to the change in temperature. Heat is being added or taken away. C) A slanted line or curved line on the graph usually indicates a type of Isothermic change (No change in Temperature), thus the change in volume is inversely proportional to the change in pressure. There is no change in heat, but work is being done by the gas or on the gas. D) A graph that has a closed shape (i.e. a triangle or similar shape) that there is a net increase in the heat of the system, the area of the shape is the energy change.