Mechanics Formulas Reference

Kinematics Formulas (Constant Acceleration)

v = v_0 + at connects final velocity to initial velocity, acceleration, and time. This is the most basic kinematics equation and is used whenever time is known or needed. It tells you the velocity at any instant during uniformly accelerated motion.

x = v_0 t + one half a t squared gives displacement from initial velocity, time, and acceleration. This equation is used when you need to find how far an object has traveled during a given time interval, or when you need to find the time to reach a certain position.

v squared = v_0 squared + 2ax relates velocities and displacement without involving time. This is the equation of choice when time is neither given nor required. It is particularly useful for finding final speed after a known displacement, or finding the displacement needed to reach a certain speed.

x = one half (v_0 + v) t gives displacement from the average of initial and final velocities multiplied by time. This equation is convenient when both velocities are known and you need the displacement, or when displacement and time are known and you need to find the other velocity.

For free fall near Earth's surface, use a = g = 9.8 m/s squared directed downward. For projectile motion, apply these equations independently to horizontal (a_x = 0) and vertical (a_y = minus g) components. The horizontal and vertical motions are independent, which is why projectile motion problems decompose into two separate kinematics problems.

Newton's Laws and Force Formulas

Newton's second law: F_net = ma. This is the central equation of classical mechanics. The net force (vector sum of all forces) on an object equals its mass times its acceleration. Use this equation whenever you need to find an unknown force, mass, or acceleration.

Weight: W = mg, the gravitational force on an object near Earth's surface. Normal force on a flat surface: N = mg (when no other vertical forces act). Normal force on an incline of angle theta: N = mg cos(theta). These normal force formulas assume the object is in equilibrium perpendicular to the surface.

Friction: f_s is less than or equal to mu_s N for static friction (self-adjusting up to a maximum), and f_k = mu_k N for kinetic friction (constant value once sliding begins). The coefficient of static friction mu_s is typically larger than the kinetic coefficient mu_k for the same pair of surfaces.

Spring force (Hooke's law): F = minus kx, where k is the spring constant and x is the displacement from equilibrium. The negative sign indicates the force always opposes the displacement, pulling the object back toward the equilibrium position.

Centripetal acceleration: a_c = v squared / r, always directed toward the center of the circular path. Centripetal force: F_c = mv squared / r, which is not a new force but the name for whatever real force provides the inward acceleration. Gravitational force between two masses: F = G m1 m2 / r squared, where G = 6.674 times 10 to the minus 11 N m squared / kg squared.

Energy Formulas

Kinetic energy: KE = one half mv squared. This is the energy an object has due to its motion. Note the velocity-squared dependence: doubling the speed quadruples the kinetic energy, which is why stopping distance increases so dramatically at higher speeds.

Gravitational potential energy near a surface: PE = mgh, where h is the height above a chosen reference point. The reference point is arbitrary; only differences in PE matter physically. Elastic potential energy: PE = one half kx squared, where k is the spring constant and x is the displacement from natural length.

Work: W = Fd cos(theta), where theta is the angle between the force and the displacement. Work is positive when force and displacement are in the same direction, negative when they oppose each other, and zero when they are perpendicular. The work-energy theorem states W_net = delta KE, connecting work to changes in speed.

Power: P = W/t (average power) or P = Fv (instantaneous power). Power is measured in watts (1 W = 1 J/s). Conservation of mechanical energy (no friction): KE_i + PE_i = KE_f + PE_f. With friction present: KE_i + PE_i = KE_f + PE_f + energy lost to friction.

For objects far from Earth: general gravitational PE = minus G m1 m2 / r. Escape velocity: v_esc = square root of (2GM/r), the minimum speed needed to escape a gravitational field without further propulsion.

Momentum and Impulse Formulas

Linear momentum: p = mv, a vector quantity. Impulse: J = F delta t = delta p, connecting force and time to the change in momentum. This is Newton's second law in its original form and explains why extending collision time (airbags, crumple zones) reduces peak force.

Conservation of momentum for a two-object system: m1 v1_i + m2 v2_i = m1 v1_f + m2 v2_f. This applies in every direction independently. For perfectly inelastic collisions (objects stick together): m1 v1_i + m2 v2_i = (m1 + m2) v_f. For elastic collisions between equal masses in one dimension, the objects simply exchange velocities.

Center of mass position: x_cm = (m1 x1 + m2 x2) / (m1 + m2). Center of mass velocity: v_cm = (m1 v1 + m2 v2) / (m1 + m2). The total momentum of a system equals the total mass times the center of mass velocity: p_total = M v_cm.

Rotational Motion Formulas

Angular velocity: omega = delta theta / delta t, measured in radians per second. Angular acceleration: alpha = delta omega / delta t, measured in radians per second squared. The relationship between linear and angular quantities: v = omega r (linear speed from angular velocity) and a_tangential = alpha r.

Rotational kinematics mirrors translational kinematics: omega = omega_0 + alpha t; theta = omega_0 t + one half alpha t squared; omega squared = omega_0 squared + 2 alpha theta. These apply only when angular acceleration is constant, just as their translational counterparts apply only when linear acceleration is constant.

Torque: tau = rF sin(theta), where r is the distance from the axis and theta is the angle between force and lever arm. Newton's second law for rotation: tau_net = I alpha. Angular momentum: L = I omega. Conservation of angular momentum: I_1 omega_1 = I_2 omega_2 when no external torque acts.

Rotational kinetic energy: KE_rot = one half I omega squared. For rolling without slipping, the constraint v_cm = omega R links translational and rotational motion. The parallel axis theorem: I = I_cm + Md squared allows calculation of moment of inertia about any axis parallel to one through the center of mass.

Common moments of inertia: solid cylinder about its central axis I = one half MR squared; hollow cylinder I = MR squared; solid sphere I = two fifths MR squared; hollow sphere I = two thirds MR squared; thin rod about center I = one twelfth ML squared; thin rod about end I = one third ML squared; thin hoop I = MR squared.

Simple Harmonic Motion Formulas

Position as a function of time: x(t) = A cos(omega t + phi), where A is amplitude, omega is angular frequency, and phi is the phase constant. Angular frequency for a mass-spring system: omega = square root of (k/m). For a simple pendulum (small angle): omega = square root of (g/L).

Period: T = 2 pi / omega. Frequency: f = 1/T = omega / (2 pi). Maximum speed: v_max = A omega (occurs at equilibrium). Maximum acceleration: a_max = A omega squared (occurs at extremes). Total energy: E = one half kA squared (constant throughout the motion in the absence of damping).

Fluid Mechanics Formulas

Pressure: P = F/A, measured in pascals (Pa). Pressure at depth h in a fluid: P = P_0 + rho g h, where P_0 is the surface pressure and rho is the fluid density. This equation explains why pressure increases with depth and why dams must be stronger at their bases.

Buoyant force (Archimedes' principle): F_b = rho_fluid times V_displaced times g. An object floats when the buoyant force equals its weight, which happens when the object's average density is less than the fluid's density. Continuity equation for incompressible flow: A1 v1 = A2 v2, stating that flow speed increases when the pipe narrows.

Bernoulli's equation: P + one half rho v squared + rho g h = constant along a streamline. This is conservation of energy applied to fluid flow. It explains why faster-moving fluids exert lower pressure and is the basis for understanding airplane lift, Venturi tubes, and many other fluid phenomena.

Tips for Using Formulas Effectively

Always solve for the unknown algebraically before substituting numbers. This reduces arithmetic errors, reveals how the answer depends on each variable, and makes it easy to check limiting cases. For example, if your formula for acceleration on an incline gives a = g sin(theta), you can verify it gives a = 0 for a flat surface (theta = 0) and a = g for a vertical drop (theta = 90 degrees).

Check that your answer has the correct units at every step. Dimensional analysis catches most algebraic mistakes. If you are calculating a force and your intermediate result has units of meters per second, you know something went wrong. Every term added or subtracted must have the same units, and every formula must be dimensionally consistent.

Know when each formula applies. Kinematics equations require constant acceleration. Conservation of mechanical energy requires no friction (or you must account for friction work separately). Momentum conservation requires negligible external forces during the interaction. Applying a formula outside its valid conditions gives a wrong answer regardless of how carefully you do the math.

When facing a complex problem, start by identifying which branch of mechanics applies (kinematics, forces, energy, momentum, rotation), then select the appropriate formulas. Many problems can be solved by multiple methods, and choosing the most efficient one saves time. Energy methods avoid the need to track accelerations. Momentum methods avoid the need to know forces during brief collisions. Experience teaches which approach is fastest for each problem type.