AP Physics C Formula Derivations Flashcards

Constant Speed and Velocity

• Constant speed: s = \frac{d}{t}

  • s: speed (m/s)

  • d: distance (m)

  • t: time (s)

• Constant velocity: v = \frac{\Delta x}{t}

  • v: velocity

  • \Delta x: displacement

  • t: time

Acceleration

• Acceleration: a = \frac{\Delta v}{t}

  • a: acceleration (m/s²)

  • \Delta v: change in velocity (m/s)

  • t: time (s)

Trigonometry Identity

• 2\sin{\Theta}\cos{\Theta} = \sin{2\Theta}

Kinematics with Constant Acceleration

• vf = vo + at

• vf^2 = vo^2 + 2a\Delta x

  • v_f: final velocity (m/s)

  • v_o: initial velocity (m/s)

  • a: acceleration (m/s²)

  • t: time (s)

  • \Delta x: displacement (m)

• Vertical motion with constant gravity can be analyzed by replacing x with y.

• \Delta x = v_o t + \frac{1}{2} a t^2

• \Delta x = \frac{1}{2} (vo + vf) t

Integrals in Kinematics

• Displacement as the integral of velocity: \Delta x = \int v(t) dt

  • \Delta x: displacement (m)

  • v(t): velocity as a function of time (m/s)

  • t: time (s)

• Change in velocity as the integral of acceleration: \Delta v = \int a(t) dt

  • \Delta v: change in velocity (m/s)

  • a(t): acceleration as a function of time (m/s²)

  • t: time (s)

Forces and Acceleration

• Newton's Second Law: \Sigma F = ma

  • \Sigma F : Sum of forces in one dimension (N)

  • m: mass (kg)

  • a: acceleration (m/s²)

• Weight: w = mg

  • w: weight (N)

  • m: mass (kg)

  • g: gravity (m/s²)

Friction

• Friction force: Ff=\mu F_{N}

  • F_f : Friction force (N)

  • \mu: Coefficient of friction (no unit)

  • F_N : Normal force (N)

• Normal force with an angled pull: F_N = mg \pm \sin{\Theta}

Incline

• Component of weight parallel to the incline: w_x = \sin{\theta} mg

  • w_x: Component of weight parallel to the incline (N)

  • \theta: Angle of incline (°)

  • m: mass (kg)

  • g: gravity (m/s²)

• Component of weight perpendicular to the incline (Normal Force): F_N = \cos{\theta} mg

  • F_N: Normal force (N)

  • \theta: Angle of incline (°)

  • m: mass (kg)

  • g: gravity (m/s²)

Static Equilibrium

• Sum of horizontal forces: \Sigma F_x = 0

• Sum of vertical forces: \Sigma F_y = 0

• Sum of torques: \Sigma \tau = 0

Torque

• Torque: \tau = r \times F or \tau = Fd\sin{\theta}

  • \tau: torque (Nm)

  • F: force (N)

  • r: distance from axis (m)

  • d: perpendicular distance from axis (m)

Work

• Work: W = Fd\cos{\theta}

  • W: work (J)

  • F: force (N)

  • d: distance (parallel to force) (m)

• Work as the integral of force and distance: W = \int F dr

  • W: work (J)

  • F: force (N)

  • r: distance (m)

Kinetic Energy

• Kinetic energy: KE = \frac{1}{2} mv^2

  • KE: kinetic energy (J)

  • m: mass (kg)

  • v: velocity (m/s)

Work-Kinetic Energy Theorem

• \Delta KE = \Sigma W = \Sigma F_{|} d

  • \Delta KE: change in KE (J)

  • \Sigma W: net work (J)

  • F_{|}: parallel force (N)

  • d: distance (m)

Gravitational Potential Energy

• Gravitational potential energy: U_g = mgh

Potential Energy and Force

Change in potential energy is the negative integral of force and distance: \Delta U = -\int F(r) dr
* \Delta U: change in potential energy (J)
* F: force (N)
* r: distance (m)

• Force is the negative derivative of potential energy with respect to distance: F = -\frac{dU(x)}{dx}

  • F: force (N)

  • U: potential energy (J)

  • x: distance (m)

Spring Potential Energy

• Spring potential energy: U_s = \frac{1}{2} kx^2

  • U_s: potential energy (J)

  • k: spring constant (N/m)

  • x: position from equilibrium (m)

Non-Conservative Work

• Work done by non-conservative forces: W{NC} = F{NC} d

  • W_{NC}: Work done by non-conservative forces (J)

  • F_{NC}: Non-conservative force (friction, air resistance) (N)

  • d: Distance (m)

Conservation of Energy

• Conservation of mechanical energy: E{MECH} = PEg + PEs + KE ; E{Ti} = E_{Tf}

  • E_{MECH}: Total mechanical energy (J)

  • PE_g: Gravitational Potential Energy (J)

  • PE_s: Spring Potential Energy (J)

  • KE: Kinetic Energy (J)

  • E_{Ti}: Initial total energy (J)

  • E_{Tf}: Final total energy (J)

• Conservation of total energy: ET = PEg + PEs + KE + W{NC}

  • E_T: total energy (J)

  • PE_g: potential energy (J)

  • PE_s: potential energy (J)

  • KE: kinetic energy (J)

  • W_{NC}: work done by non-conservative forces (J)

Spring Force

• Spring force: F = -k\Delta x

  • F: spring force (N)

  • k: spring constant (N/m)

  • \Delta x: position from equilibrium (m)

Power

• Power: P = \frac{W}{t} = \frac{\Delta E_T}{t}

  • P: power (W or J/s)

  • W: work (J)

  • \Delta E_T: total change in energy (J)

  • t: time (s)

• Constant velocity power: P = Fv

  • P: power (W or J/s)

  • F: force (N)

  • v: velocity (m/s)

• Instantaneous power is the derivative of work with respect to time: P_{inst} = \frac{dW}{dt}

  • P: power (W or J/s)

  • W: work (J)

  • t: time (s)

Momentum

• Momentum: p = mv

  • p: momentum (kg m/s)

  • m: mass (kg)

  • v: velocity (m/s)

Newton's Second Law with Momentum

• \Sigma F = \frac{\Delta p}{t}

  • \Sigma F: force (N)

  • \Delta p: change in momentum (Ns)

  • t: time (s)

• Force is the derivative of momentum with respect to time: F_{net} = \frac{dp}{dt}

  • F: force (N)

  • p: momentum (Ns)

  • t: time (s)

Impulse

• Impulse is the integral of force with respect to time: J = \int F_{net}(t) dt = \Delta p

  • J: impulse (Ns)

  • F: force (N)

  • t: time (s)

  • \Delta p: change in momentum (kg*m/s or Ns)

Collisions

• Inelastic collision: m1v1 + m2v2 = m1v1' + m2v2'

  • m_1: mass 1 (kg)

  • v_1: velocity 1 (m/s)

  • m_2: mass 2 (kg)

  • v_2: velocity 2 (m/s)

  • v_1': velocity 1 after collision (m/s)

  • v_2': velocity 2 after collision (m/s)

• Elastic collision: v1 - v2 = v2' - v1'

• Perfectly inelastic collision: m1v1 + m2v2 = (m1 + m2)v_f

  • m_1: mass 1 (kg)

  • v_1: velocity 1 (m/s)

  • m_2: mass 2 (kg)

  • v_2: velocity 2 (m/s)

  • v_f: final velocity of combined mass (m/s)

• Recoil: m{total}vo = m1v{1f} + m2v{2f}

Pendulum Height

• Finding height for a pendulum at an angle: h = L - \cos{\theta}L

  • h: height above equilibrium (m)

  • L: length of pendulum (m)

  • \theta: angle (°)

Center of Mass

• Center of mass (horizontal): x_{cm} = \frac{\Sigma mx}{\Sigma m}

  • x_{cm}: horizontal center of mass (m)

  • m: mass (kg)

  • x: position from reference point (m)
    Vertical center of mass can be found by replacing x with y.

Velocity of Center of Mass

• v_{cm} = \frac{\Sigma p}{\Sigma m} = \frac{\Sigma mv}{\Sigma m}

  • v_{cm}: velocity of center of mass (m/s)

  • \Sigma p: sum of momentum (kg m/s)

  • \Sigma m: sum of mass (kg)

  • m: mass of particle (kg)

  • v: velocity of particle (m/s)

Linear Mass Density

• \lambda = \frac{dm}{dl} OR dm = \lambda dx

  • \lambda: linear mass density (kg/m)

  • dm: mass (kg)

  • dl: length (m)

Center of Mass Integration

• r_{cm} = \frac{\int r dm}{\int dm} OR \frac{1}{M} \int x dm

  • r_{cm}: center of mass (m)

  • r: distance (m)

  • M: total mass (kg)

Tangential Velocity

• Tangential velocity of a circular orbit: v_T = \frac{2\pi r}{T}

  • v_T: tangential velocity (m/s)

  • r: radius of orbit (m)

  • T: period of orbit (s)

Centripetal Acceleration

• ac = \frac{vT^2}{r} = r\omega^2

  • a_c: centripetal acceleration (m/s²)

  • v_T: tangential velocity (m/s)

  • r: radius (m)

  • \omega: angular velocity (rad/s)

Centripetal Force

• Fc = m \frac{vT^2}{r} or Fc = mac or F_c = m \omega^2 r

  • F_c: centripetal force (N)

  • m: mass (kg)

  • v_T: tangential velocity (m/s)

  • r: radius (m)

  • a_c: centripetal acceleration (m/s²)

  • \omega: angular velocity (rad/s)

Circular motion Scenarios

• Car going around a flat turn: Ff = Fc

  • F_f: Friction force (N)

  • F_c: Centripetal force (N)

• Car going around a banked turn (no friction): FN\sin{\Theta} = \frac{mvT^2}{r}

  • F_N: normal force (N)

  • m: mass (kg)

  • v_T: velocity (m/s)

  • r: radius (m)

Vertical Loops

• Bottom of a vertical loop: Fc = FN - w ; FN = Fc + w

• Top of a vertical loop (upside down): Fc = FN + w ; FN = Fc - w

  • F_N: normal force or apparent weight (N)

  • F_c: centripetal force (N)

  • w: weight (N)

• Top of a vertical loop (right side up): Fc = w - FN ; FN = w - Fc

• Top of a vertical loop (weightless): Fc = w ; ac = g ; v = \sqrt{rg}

Apparent Gravity

• Apparent gravity: g{app} = g \pm a{ext}

  • g_{app}: apparent gravity (m/s²)

  • g: gravity (m/s²)

  • a_{ext}: external acceleration (m/s²)

• Apparent weight: FN = mg \pm ma{ext}

  • F_N: apparent weight (N)

  • m: mass (kg)

  • g: gravity (m/s²)

  • a_{ext}: external force (N)

Gravitational Force

• F = G \frac{m1 m2}{r^2}

  • F: Gravitational force (N)

  • G: Gravitational constant (Nm²/kg²)

  • m_1: mass (kg)

  • m_2: mass (kg)

  • r: radius (m)

Gravity

• g = G \frac{m_p}{r^2}

  • g: gravity (m/s²)

  • G: Gravitational constant (Nm²/kg²)

  • m_p: mass of planet (kg)

  • r: radius (m)

Kepler's Third Law

• T^2 = (\frac{4\pi^2}{GM})r^3

  • T: period of orbit (s)

  • G: grav. constant (Nm²/kg²)

  • M: mass of object being orbited (kg)

  • r: radius of orbit (m)

Circular Orbit

• a_c = g

  • a_c: centripetal acceleration (m/s²)

  • g: gravity (m/s²)

• Speed of a circular orbit: v = \sqrt{\frac{GM}{r}}

  • v: velocity of satellite (m/s)

  • G: grav. constant (Nm²/kg²)

  • M: mass of object being orbited (kg)

  • r: radius of orbit (m)

Angular to Linear Conversion

• Distance: \Delta x = r\theta

  • \Delta x: linear distance (m)

  • r: radius of rotation (m)

  • \theta: angular displacement (rad)

• Velocity: v_T = r\omega

  • v_T: linear (tangential) velocity (m/s)

  • r: radius of rotation (m)

  • \omega: angular velocity (rad/s)

• Acceleration: a_T = r\alpha

  • a_T: linear acceleration (m/s²)

  • r: radius of rotation (m)

  • \alpha: angular acceleration (rad/s²)

• Angular velocity: \omega = 2\pi f or T = \frac{2\pi}{\omega}

  • \omega: angular velocity (rad/s)

  • f: frequency (Hz)

  • T: period (s)

• Angular velocity is the derivative of angular displacement with respect to time: \omega = \frac{d\theta}{dt}

  • \omega: angular velocity (rad/s)

  • \theta: angular displacement (rad)

  • t: time (s)

• Angular acceleration is the derivative of angular velocity with respect to time: \alpha = \frac{d\omega}{dt}

  • \alpha: angular acceleration (rad/s²)

  • \omega: change in angular velocity (rad/s)

  • t: time (s)

Rotational Kinematics

• \omegaf = \omegao + \alpha t

  • \omega_f: final angular velocity (rad/s)

  • \omega_o: initial angular velocity (rad/s)

  • \alpha: angular acceleration (rad/s²)

  • t: time (s)

• \omegaf^2 = \omegao^2 + 2\alpha \Delta \theta

• \Delta \theta = \omega_o t + \frac{1}{2} \alpha t^2

• \Delta \theta = \frac{1}{2} (\omegao + \omegaf) t

  • \Delta \theta: angular displacement (rad)

Rotational Inertia

• General formula: I_{tot} = \Sigma I = \Sigma mr^2

  • I: rotational inertia

  • m: mass (kg)

  • r: radius distance from pivot (m)

• Inertia of a particle: I = mr^2

  • m: mass (kg)

  • r: distance from pivot (m)

• Inertia of a solid disc (cylinder): I = \frac{1}{2} mr^2

  • m: mass (kg)

  • r: radius of disc (m)

• Inertia of a hollow cylinder (hoop): I = mr^2

  • m: mass (kg)

  • r: radius of cylinder (m)

• Inertia of a solid sphere: I = \frac{2}{5} mr^2

  • m: mass (kg)

  • r: radius of sphere (m)

• Inertia of a rod (pivot in the middle): I = \frac{1}{12} mL^2

  • m: mass (kg)

  • L: length of rod (m)

• Inertia integration: I = \int r^2 dm

• Parallel axis theorem: I = I_{CM} + Md^2

  • I: inertia at pivot (kg m²)

  • I_{CM}: inertia at center of mass (kg m²)

  • M: mass (kg)

  • d: distance from pivot to center of mass (m)

Rotational Kinetic Energy

• KE_R = \frac{1}{2} I\omega^2

  • KE_R: rotational KE (J)

  • I: rotational inertia (kg m²)

  • \omega: angular velocity (rad/s)

Overall Acceleration

• For a rotating particle: a = \sqrt{ac^2 + aT^2}

  • a: overall accel. (m/s²)

  • a_c: centripetal accel. (m/s²)

  • a_T: tangential accel. (m/s²)

Rotational Work

• W = \tau \Delta \Theta

• Work is the integral of torque and angular displacement: W = \int \tau d\Theta

  • W: Work done (J)

  • \tau: external torque (Nm)

  • \Delta \Theta: angular displacement (rad)

Conservation of Energy with Rotation

• ET = PEg + KER + KE + W{\tau}

  • E_T: total energy (J)

  • PE_g: gravitational potential energy (J)

  • KE_R: rotational kinetic energy (J)

  • KE: translational kinetic energy (J)

  • W_{\tau}: work done by external torque (J)

External Torque Creating Angular Acceleration

• \Sigma \tau_{ext} = I\alpha OR \alpha = \frac{\Sigma \tau}{I}

  • \Sigma \tau_{ext}: sum of external torque (Nm)

  • I: rotational inertia (kg*m²)

  • \alpha: angular acceleration (rad/s²)

• \Sigma \tau_{ext} = \frac{\Delta L}{t}

  • \Sigma \tau: sum of torque (Nm)

  • \Delta L: change in angular momentum (kg m²/s)

  • t: time (s)

Angular Momentum

• L = r \times p = I\omega

  • L: angular momentum (kg m²/s)

  • r: radius (m)

  • p: linear momentum (kg*m/s)

  • I: rotational inertia (kg*m²)

  • \omega: angular velocity (rad/s)

• Change in angular momentum is the integral of external torque with respect to time: \Delta L = \int \tau dt

  • \Delta L: change in angular momentum (kg m²/s)

  • \tau: external torque (Nm)

  • t: time (s)

• Angular momentum of a particle: L = mvr\sin{\Theta}

  • L: angular momentum (kg m² / s)

  • m: mass (kg)

  • v: velocity (m/s)

  • r: radius (m)

Collisions Creating Rotation

• Elastic / Inelastic: I1\omega1 + I2\omega2 = I1\omega1' + I2\omega2'

  • I: rotational inertia (kg m²)

  • \omega: angular velocity (rad/sec)

• Perfectly Inelastic: I1\omega1 + I2\omega2 = (\Sigma I)\omega_f

  • I: rotational inertia (kg m²)

  • \omega: angular velocity (rad/sec)

Gravitational Potential Energy

• U_G = -G \frac{Mm}{R}

  • U_G: potential energy (J)

  • G: gravitational constant (Nm²/kg²)

  • M: mass of planet (kg)

  • m: mass of satellite (kg)

  • R: radius (m)

Total Energy

• Circular Orbit: E_T = -G \frac{Mm}{2R}

  • E_T: total energy (J)

  • G: gravitational constant (Nm²/kg²)

  • M: mass of planet (kg)

  • m: mass of satellite (kg)

  • R: radius (m)

• Elliptical Orbit: E_T = -G \frac{Mm}{2a}

• E_T = \frac{1}{2}mv^2 – G \frac{Mm}{r}

  • E_T: total energy (J)

  • G: grav. cons. (Nm²/kg²)

  • M: mass of planet (kg)

  • m: mass of satellite (kg)

  • a: semi-major axis (m)

• Semi-major axis: a = \frac{apogee + perigee}{2}

  • a: semi-major axis (m)

Conservation of Momentum (Elliptical Orbit)

• mv1r1 = mv2r2 ; v1r1 = v2r2

  • m: mass of satellite (kg)

  • v: velocity of satellite (m/s)

  • r: radius of orbit (m)

Escape Speed

• v = \sqrt{\frac{2GM}{r}}

  • v: escape velocity (m/s)

  • G: gravitational constant (Nm²/kg²)

  • M: mass of planet (kg)

  • r: radius of launch (m)

Period and Frequency

• T = \frac{1}{f}

  • T: period (s)

  • f: frequency (Hz)

Pendulums

• Simple Pendulum: T = 2\pi \sqrt{\frac{L}{g}}

  • T: period (second (s))

  • L: length of pendulum (meter (m))

  • g: gravity (m/s²)

• Mass-Spring System: T = 2\pi \sqrt{\frac{m}{k}}

  • T: period (second (s))

  • m: mass (kg)

  • k: spring constant (Newton/meter (N/m))

• Physical Pendulum: T = 2\pi \sqrt{\frac{I}{mgd}}

  • T: period (sec)

  • I: inertia (kg*m²)

  • m: mass (kg)

  • g: gravity (m/s²)

  • d: distance from pivot to center of mass (m)

• Torsional Pendulum: T = 2\pi \sqrt{\frac{I}{k}}

  • T: period (sec)

  • I: inertia (kg*m²)

  • k: torsional constant (kg*m²/s²)

Mass-Spring System Equations

• Position as a function of time: x(t) = A \cos{2\pi ft} OR x = x_{max} \cos(\omega t +\varphi)

  • x(t): position from equilibrium (m)

  • A: amplitude (m)

  • f: frequency (Hz)

  • t: given time (s)

  • \omega: angular velocity (rad/s)

  • \varphi: phase angle (radian)

• Velocity as a function of time: v(t) = -v_{max} \sin{2\pi ft}

Maximum Values

• Maximum velocity: v_{max} = 2\pi f A

  • v_{max}: maximum velocity (m/s)

  • f: frequency (Hertz (Hz))

  • A: amplitude (meter (m))

• Acceleration as a function of time: a(t) = -a_{max} \cos{2\pi ft}

• Maximum acceleration: a_{max} = (2\pi f)^2 A

  • a_{max}: maximum acceleration (meters/second² (m/s²))

  • f: frequency (Hertz (Hz))

  • A: amplitude (meter (m))

• a_{max} = -\frac{kA}{m}

Springs Connected

• In Parallel: k_T = \Sigma k

• In Series: \frac{1}{k_T} = \Sigma \frac{1}{k}

Graph Analysis

• Position vs. Time

  • Axis labels: x (m) vs. t (s)

  • Slope (Derivative): velocity (m/s)

• Velocity vs. Time

  • Axis labels: v (m/s) vs. t (s)

  • Slope (Derivative): acceleration (m/s²)

  • Area (Integral): displacement (m)

• Acceleration vs. Time

  • Axis labels: a (m/s²) vs. t (s)

  • Area (Integral): change in velocity (m/s)

• Drag Force / Weight vs. v_T^2

  • Axis labels: Force (N) vs. terminal vel.² (m²/s²)

  • Slope: drag coefficient

• Force vs. Distance

  • Axis labels: F (N) vs. d (m)

  • Slope: k (N/m) (only if it’s a spring)

  • Area (Integral): work/energy/\Delta KE/\Delta PE (J)

• Kinetic Energy / Work vs. Distance

  • Axis labels: KE/W (J) vs. d (m)

  • Slope: force (N)

• Potential Energy vs. Distance

  • Axis labels: PE (J) vs. d (m)

  • Slope: negative force (N)

• \Delta Energy / Work vs. Time

  • Axis labels: KE/PE/W (J) vs. t (s)

  • Slope: power (W)

• Force vs. Time

  • Axis labels: F (N) vs. t (s)

  • Area (Integral): impulse/change in momentum (\Deltap) (kg m/s)

• \Delta Momentum / Impulse vs. Time

  • Axis labels: p / J (kgm/s or Ns) vs. t (s)

  • Slope: Force (N)

• Angular Displacement vs. Time

  • Axis labels: radians (\Theta) vs. t (s)

  • Slope: \omega (rad/s)

• Angular Velocity vs. Time

  • Axis labels: \omega (rad/s) vs. t (s)

  • Slope: \alpha (rad/s²)

  • Area: \Delta \Theta (radians)

• Angular Acceleration vs. Time

  • Axis labels: \alpha (rad/s²) vs. t (s)

  • Area: \Delta \omega (radians / second)

• KE_R vs. \omega^2

  • Axis labels: KE_R (J) vs. \omega^2 (rad² / sec²)

  • Slope: rotational inertia (kg*m²); I = 2 x slope

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