AP Physics C Mechanics EXAM REVIEW

created 2025; EHS

vectors

quantities with magnitude and direction; ex- velocity, displacement, acceleration, force, momentum

can be written in component form: v= ai +bj+ ck

scalar

quantities with only a magnitude; ex- distance, speed, mass, energy, temperature, time

Dot Product (vector multiplication)

method 1: A*B = |A||B|cosθ

method 2: A*B= A_xB_x + A_yB_y + A_zB_z

produces a scalar

Cross Product (vector multiplication)

method 1: |AxB| = |A||B| sinθ

method 2: if A= A_xi + A_yj + A_zk and B= B_xi + B_yj + B_zk

then AxB= (A_yB_z - A_zB_y)i - (A_xB_z + A_zB_x)j + (A_xB_y - A_yB_x)j

produces a vector

average displacement, velocity, and acceleration

displacement: ∆x=(x_f - x_i) units: m

velocity: ∆v= ∆x/t units: m/s

acceleration: ∆a= ∆v/t units: m/s²

instantaneous displacement, velocity, and acceleration

displacement: x(t)= ∫v(t) dt

velocity: v_inst= dx/dt; v(t)= ∫a(t) dt

acceleration: a_inst=dv/dt OR a_inst=d²x/dt²

graphical representation of position, velocity, and acceleration

slope of x(t) = v(t)

slope of v(t) = a(t)

area under curve of a(t) = v(t)

area under curve of v(t) = x(t)

how to linearize data in desmos

steps

• add table, insert values, and adjust window

• determine the type of non-linear relationship (ex: inverse, square, inv-square, root, inv-root, quadratic, log, exponential)

• create new variables to represent the transformation

• plot the new data points

• draw a line of best fit

non-linear power function relationship examples (c is constant):

• inverse: y= c/x

• square: y= cx²

• inv-square: y= c/x²

• root: y= c√x

• inv-root: y=c/√x

kinematic equations for constant acceleration

v_f= v_i + at

x= x_i + v_it + 1/2at²

v_f²= v_i² + 2a(x_f - x_i)

x= 1/2(v_i + v_f)t

only when acceleration is constant!!

Reference frames

the velocity of an object depends on the location and motion of the observer.

EX1: Object A is moving left at 4m/s and Object B is moving is moving right at 6 m/s. the velocity of Object A from the reference frame of Object B is 10 m/s to the left.

EX2: an observer on land is watching a river. the velocity of the water relative to the observer is V_WL, the velocity of the boat relative to the water is V_BW, the boat moves perpendicular to V_WL. what is the velocity of the boat relative to the land? V_BL² = V_BW² + V_WL²

projectile motion

break down into horizontal and vertical components of d, v, and a. a faster horizontal velocity will result in a greater horizontal distance. horizontal motion of an object does not affect its vertical motion. the horizontal motion of a projectile is independent of its vertical motion and the vertical motion is independent of its horizontal motion.

the horizontal acceleration is zero and the vertical acceleration is constant. use kinematic equations in the vertical and horizontal directions separately. the time of the horizontal and vertical directions is the same.

launch angles in projectile motion

the greater the launch angle the greater the vertical distance of the projectile. the optimal angle for max range is 45 degrees.

projectile motion equations

h_max= (v_i²sin²θ_i)/(2g)

t_{max h}= (v_isinθ_i)/g

R= (v_i²sin²θ_i)/g

R_max=v_i²/g

Uniform Circular Motion

a_c= v²/r

v= rω

T= (2πr)/v

ω= (2π)/T

a_c= rω²

Tangential and Radial Acceleration

a= √(a_r²+a_t²)

QUESTION: which of the following are accelerating?

1. a car slowing down

2. ball being swung in a circle

3. vibrating string

4. moon orbiting

5. skydiver at terminal speed

6. astronaut orbiting in space

7. ball rolling down a hill

8. person driving straight at a constant speed with their foot on the gas

objects changing direction or changing speed are accelerating. therefore 1, 2, 3, 4, 6, and 7 are accelerating while 5 and 8 are not

newtons first law

in the absence of external forces and when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion will stay in motion with a straight-line velocity, unless acted on by an external force.

-condition 1: object can move but must be at a constant speed

-condition 2: object is a rest

constraint: forces must be balanced, sum of forces is zero

inertial frame of reference

a non-accelerating frame of reference

inertia vs weight

inertia- quantity of matter, aka mass (unit kg)

weight- mass effected by gravity; w=mg; (unit newton)

mass never changes; weight changes with gravity

facts about forces

unit: Newton (N)

a push or a pull

can exist during physical contact or without contact

equilibrium

when the sum of the forces acting on the object is zero; F_net=0

newtons second law

the acceleration of an object is directly proportional to the net force and inversely proportional to the mass;

a=F_net/m

ΣF=ma

static vs kinetic friction

static- friction that keeps an object at rest and prevents it from moving

F_f=F_Nµ_s

kinetic- friction that acts during motion

F_f=F_Nµ_k

µ has no units

friction depends on the materials sliding against each other, NOT surface area!

newtons third law

for every action force there is an equal and opposite reaction force

• they NEVER cancel out

ie: a person exerts a force on a wall and the wall exerts a force on the person

ie: m_car A_car = M_train a_train

Air resistance

linear: R= -bv

• b is a constant

• velocity (v) is changing

m(dv/dt)=mg-bv

to find terminal 0=mg-bv therefore v_T=mg/b

centripetal force

F_c=mv²/r

has to be supplied by another force

ie: a cart on a roller coaster at the top of a hill the F_c=F_g-F_N while a cart at the bottom of the hill F_c=F_N-F_g

centripetal force is the net force directed toward the center of a circular path, allowing an object to maintain circular motion. it can be provided by tension, gravity, friction, depending on the situation.

resistive forces

R=-bv

basics of energy

• work- force applied through a distance; must be parallel; W=Fd

• kinetic energy- energy of motion; KE= ½ mv²

• potential energy- energy of position; 2 types: gravitational (U= mgh) and elastic (U= ½ kx²)

• power- rate at which energy is transferred/consumed- P=W/t or P=E/t

work

W= F∆d → FdCosθ

work is found by multiplying the force times the displacement and result is energy.

units: joules (J)

F and d must be parallel

W= ∫ F(x)dx → work is the area under the force vs displacement graph

W_net= ∆K → W=(mv²/2 - mv₀²/2)

W= -∆U → W= mgy - mgy₀ → work equals the negative change in potential energy

energy conservation

∆K = -∆U

K-K₀ = -(U-U₀₎

in a closed isolated system energy is conserved, meaning the initial energy equals the final energy.

Elastic Potential Energy

hooks law: F_s=-Kx

negative because it’s a restoring force that works in the opposite direction of the displacement

U_s= ½ Kx²

power

Power = work/time

P= Energy/time

P= F v → P= F(d/t)

Momentum and Impulse

• momentum: moving mass- inertia in motion

  • momentum = mass x velocty →(p= mv); units: kgm/s

• Impulse: applying a force for a period of time to change momentum

  • Impulse= force_avg x time →(J= F_avgt) or J= ∆p; units: Ns or Kgm/s

• relationship: the area under the curve of a force vs time graph is the impulse/change in momentum.

• increasing momentum: apply the greatest force for the longest period of time (ie: following through)

• Decreasing momentum: extending the impact time (ie: air bags)

• bouncing delivers a greater impulse (∆p= 2mv)

conservation of momentum

in a closed isolated system total momentum is conserved

collisions

elastic collisions: momentum AND kinetic energy are conserved (ie: billiard balls)

inelastic collisions: ONLY momentum is conserved (ie: car crashes)

• perfectly inelastic collisions: objects stick together

• imperfectly inelastic collisions: objects don’t stick together

• pay attention to signs!

center of mass

use for elastic collisions only!

v_cm= Σ m_iv_i / m_i

subtract V_cm from initial velocity to enter Center of mass reference frame; add V_cm to final velocity to leave the center of mass reference frame.

rotation of a rigid object a fixed axis

• angular position: θ = S/r

  • S is arc length

  • r is radius of rotation

  • θ is the angle through which the object rotates

• angular speed: ω_avg = ∆θ/∆t or ω= dθ/dt or ω= 2π/T

  • ω is angular speed (units rad/sec)

  • T is the period

• Angular acceleration: α_avg= ∆ω/∆t or α= dω/dt

  • α is angular acceleration (units rad/sec²

rigid object under constant acceleration

ωf=ωi+αt

θf= θi+ωit+1/2 αt²

ωf²=ωi²+2α(θf-θi)

θf=θi+1/2(ωi+ωf)t

angular and translational quantities

v= rω

a= rα

a_c = rω²

angular speed radius does not matter

linear/translational speed radius matters

Torque

_T= rFsinθ

force should be perpendicular to the lever arm

units: Nm

Σ_T= Iα

moment of inertia

I= Σ m_ir_i²

Hoop/Thin Cylindrical Shell: I_cm= 1MR²

Solid Cylinder/disk: I_cm= 1/2MR²

Hollow cylinder: I_cm=1/2M(R₁²+R₂²)

Long thin rod with rotation axis through center: I_cm=1/12ML²

Long thin rod with rotation axis through end: I=1/3ML²

Rectangular plate: I_cm=1/12M(a²+b²)

Solid Sphere: I_cm=2/5MR²

thin spherical shell: I_cm= 2/3MR²

Parallel Axis Theorem

I= I_cm+ MD²

• D= distance between parallel lines

Rotational Kinetic Energy

K_R=1/2 Iω²

energy considerations

dW= Fds = (F sinθ)rdθ

Power= dW/dt = tω

rolling motion of a rigid object

V_T= ωR

V_CM= Rω

a_cm= Rα

angular momentum

L= position cross linear momentum

L= mvr sinθ

L= Iω

angular momentum is conserved in an isolated system

∆L= 0 with no external torques

gravity

F_g= G(m₁m₂)/r²

don’t forget r is measured to the center of mass!!

U_g= Gm₁m₂/r

escape speed as the particle gets infinitely faraway velocity approaches 0

oscillatory motion

F_s= -kx

a_x=-k/m x

ω=√(k/m)

x(t)= Acos(ωt+∅)

Vmax= ωA

a(max)= ω²A

spring constant (k)

in series: k_total= (1/k)+(1/k)…

in parallel: k_total= k+k+k…