Fundamental Physics Equations

Why This Matters

Physics equations aren't just formulas to memorize—they're the language that describes how and why objects move, interact, and exchange energy. On the AP Physics 1 exam, you're being tested on your ability to connect equations to physical situations, recognize which principle applies to a given scenario, and understand the relationships between variables. The exam loves asking you to predict what happens when one quantity changes, or to explain why a particular equation applies to a situation.

These equations fall into distinct conceptual categories: kinematics (describing motion), dynamics (explaining what causes motion), energy and work (tracking transfers and transformations), momentum (analyzing collisions and interactions), and circular and oscillatory motion (specialized patterns). Don't just memorize $F = ma$—know that it's Newton's Second Law connecting net force to acceleration. When you see a collision problem, your brain should immediately think "momentum conservation" before reaching for equations. That conceptual framework is what earns you points on FRQs.

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Describing Motion: Kinematics

Kinematics equations describe how objects move without worrying about why they move. These equations assume constant acceleration and connect position, velocity, acceleration, and time. Master these relationships and you'll handle any motion problem.

Velocity-Time Relationship

• $v = v_0 + at$ connects final velocity to initial velocity, acceleration, and elapsed time
• Constant acceleration only—this equation fails the moment acceleration changes
• Graphically, this represents the slope relationship on a velocity-time graph where acceleration is the slope

Position-Time Relationship

• $x = x_0 + v_0 t + \frac{1}{2}at^2$ calculates displacement using initial conditions and constant acceleration
• Quadratic in time—the $\frac{1}{2}at^2$ term dominates at large times, creating parabolic motion paths
• Projectile motion problems use this equation separately for horizontal ($a = 0$) and vertical ($a = g$) components

Velocity-Displacement Relationship

• $v^2 = v_0^2 + 2a(x - x_0)$ eliminates time entirely, connecting velocities directly to displacement
• Time-independent—use this when the problem doesn't give or ask for time
• Energy connection—this equation is mathematically equivalent to the work-energy theorem (multiply both sides by $\frac{1}{2}m$)

Average Velocity Method

• $x = \frac{1}{2}(v + v_0)t$ uses the average of initial and final velocities to find displacement
• Only valid for constant acceleration—the average velocity trick works because velocity changes linearly
• Quick calculation tool when you already know both velocities and need displacement

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Explaining Motion: Forces and Newton's Laws

Dynamics explains why objects accelerate—forces cause changes in motion. Newton's Second Law is the bridge between forces and kinematics, while specialized force equations describe specific interactions.

Newton's Second Law

• $\vec{F}_{net} = m\vec{a}$ states that the net force on an object equals its mass times its acceleration
• Vector equation—apply separately to x and y components; forces in perpendicular directions don't affect each other
• Causal relationship—force causes acceleration; without net force, velocity stays constant (Newton's First Law)

Universal Gravitation

• $F_g = G\frac{m_1 m_2}{r^2}$ describes the gravitational attraction between any two masses
• Inverse-square law—doubling the distance reduces the force to one-quarter; this pattern appears throughout physics
• $G = 6.67 \times 10^{-11}$ N·m²/kg² is the universal gravitational constant, extremely small because gravity is the weakest fundamental force

Hooke's Law

• $F_s = -kx$ relates spring force to displacement from equilibrium, where $k$ is the spring constant
• Negative sign indicates restoring force—the spring always pushes or pulls back toward equilibrium
• Linear relationship—double the stretch, double the force; this defines an "ideal" spring

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Energy and Work: Tracking Transfers

Energy is conserved—it transforms between forms but never disappears in an isolated system. Work is the mechanism that transfers energy between objects or converts it between forms. These equations let you solve problems without knowing the details of the motion.

Kinetic Energy

• $KE = \frac{1}{2}mv^2$ represents energy of motion, depending on mass and the square of velocity
• Velocity squared—doubling speed quadruples kinetic energy; this is why car crashes at high speed are so dangerous
• Scalar quantity—direction doesn't matter; a car moving left has the same KE as one moving right at the same speed

Gravitational Potential Energy

• $PE_g = mgh$ represents stored energy due to position in a gravitational field
• Reference point matters—you choose where $h = 0$; only changes in PE have physical meaning
• Near-Earth approximation—assumes constant $g$; for planetary-scale problems, use $PE = -Gm_1m_2/r$

Elastic Potential Energy

• $PE_s = \frac{1}{2}kx^2$ represents energy stored in a compressed or stretched spring
• Displacement squared—doubling the stretch quadruples the stored energy
• Always positive—whether compressed or stretched, the spring stores energy ($x^2$ is always positive)

Work-Energy Theorem

• $W_{net} = \Delta KE$ states that net work done on an object equals its change in kinetic energy
• Work transfers energy—positive work increases KE (speeds up), negative work decreases KE (slows down)
• Connects force and energy—work equals $W = Fd\cos\theta$, where $\theta$ is the angle between force and displacement

Conservation of Mechanical Energy

• $E_i = E_f$ or $KE_i + PE_i = KE_f + PE_f$ when only conservative forces do work
• Conservative forces include gravity and springs; friction is non-conservative and removes mechanical energy
• Powerful problem-solving tool—skip the details of the path and just compare initial and final states

Power

• $P = \frac{W}{t}$ defines power as the rate of doing work or transferring energy
• Units: watts (W)—one watt equals one joule per second
• Alternative form: $P = Fv$—useful when force and velocity are known directly

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Momentum and Impulse: Collisions and Interactions

Momentum describes the "quantity of motion" and is always conserved in isolated systems. Impulse is how forces change momentum over time. These concepts dominate collision and explosion problems.

Linear Momentum

• $\vec{p} = m\vec{v}$ defines momentum as mass times velocity
• Vector quantity—direction matters; momentum in the x-direction is conserved separately from y-direction
• System property—total momentum of a system is the vector sum of all individual momenta

Impulse-Momentum Theorem

• $\vec{J} = \vec{F}_{avg}\Delta t = \Delta \vec{p}$ relates impulse (force over time) to change in momentum
• Same change, different forces—increasing collision time (like an airbag) reduces the average force
• Area under F-t curve—graphically, impulse equals the area under a force-versus-time graph

Conservation of Momentum

• $\vec{p}_{total,i} = \vec{p}_{total,f}$ states total momentum is constant when no external forces act
• Applies to all collision types—elastic, inelastic, and perfectly inelastic collisions all conserve momentum
• Explosions too—if a stationary object explodes, the pieces' momenta must sum to zero

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Circular and Rotational Motion

Circular motion requires continuous acceleration toward the center. Oscillating systems like springs and pendulums exhibit simple harmonic motion with characteristic frequencies. These specialized motions have their own vocabulary and equations.

Centripetal Acceleration

• $a_c = \frac{v^2}{r}$ describes the center-directed acceleration required for circular motion
• Always toward center—velocity is tangent to the circle, but acceleration points inward
• Not a new force—centripetal force is provided by tension, gravity, friction, or normal force; it's not a separate force

Angular Frequency of a Mass-Spring System

• $\omega = \sqrt{\frac{k}{m}}$ gives the angular frequency for simple harmonic motion of a spring-mass system
• Independent of amplitude—oscillation frequency doesn't change whether you pull the spring a little or a lot
• Period: $T = 2\pi\sqrt{\frac{m}{k}}$—larger mass means slower oscillation; stiffer spring means faster oscillation

Rotational Kinematics Connection

• $v_{cm} = r\omega$ relates linear velocity of the center of mass to angular velocity for rolling objects
• No-slip condition—this relationship holds when objects roll without slipping
• Analogous equations—rotational kinematics mirrors linear kinematics: $\omega$ replaces $v$, $\alpha$ replaces $a$, $\theta$ replaces $x$

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Torque and Rotational Equilibrium

Torque is the rotational equivalent of force—it causes angular acceleration. Objects in rotational equilibrium have zero net torque, just as objects in translational equilibrium have zero net force.

Torque

• $\tau = rF\sin\theta$ defines torque as the product of lever arm, force, and the sine of the angle between them
• Lever arm matters—the same force produces more torque when applied farther from the pivot
• Units: N·m—same units as energy, but torque and energy are different quantities

Rotational Equilibrium

• $\Sigma\tau = 0$ is the condition for rotational equilibrium—net torque equals zero
• Choose your pivot wisely—picking the pivot at an unknown force's location eliminates that force from your torque equation
• Static equilibrium requires both $\Sigma F = 0$ and $\Sigma\tau = 0$

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Quick Reference Table

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Self-Check Questions

1. Which two equations would you use together to solve a projectile motion problem where you need to find the range, and why do you apply them to different directions?

2. An object slides down a frictionless ramp. Compare solving this problem using kinematics ($v^2 = v_0^2 + 2a\Delta x$) versus energy conservation ($mgh = \frac{1}{2}mv^2$). When would you prefer each approach?

3. Two carts collide and stick together. Which quantity is definitely conserved, and which quantity is definitely not conserved? What type of collision is this?

4. A car rounds a curve at constant speed. Explain why the car is accelerating even though its speed isn't changing, and identify what force provides the centripetal acceleration.

5. Compare and contrast the roles of $F = ma$ and $\tau = I\alpha$. If an FRQ shows a wheel rolling down a ramp, which equation(s) would you need, and why might you need both?