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 ( $\Delta$ p) (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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