Ap Physics 1 Formula Sheet

Your Ultimate AP Physics 1 Formula Sheet: A Comprehensive Guide

Preparing for the AP Physics 1 exam can feel overwhelming. One of the most crucial aspects of success is mastering the fundamental formulas and their applications. This comprehensive guide serves as your ultimate AP Physics 1 formula sheet, going beyond a simple list to provide context, explanations, and examples to help you not just memorize, but truly understand these essential equations. We'll break down key concepts, explore their relationships, and offer tips for applying them effectively on the exam. This isn't just a cheat sheet; it's your roadmap to mastering AP Physics 1.

I. Kinematics: Describing Motion

Kinematics forms the foundation of classical mechanics. Understanding how objects move is crucial before tackling more complex topics. Here are the core kinematic equations:

1. Displacement:

• Δx = x_f - x_i: This defines displacement as the change in position (final position minus initial position). Remember, displacement is a vector, meaning it has both magnitude and direction.

2. Average Velocity:

• v_avg = Δx/Δt: Average velocity is the displacement divided by the change in time. Like displacement, it's a vector quantity.

3. Instantaneous Velocity:

• v = dx/dt: This represents the velocity at a specific instant in time. It's the derivative of the position function with respect to time.

4. Average Acceleration:

• a_avg = Δv/Δt: Average acceleration is the change in velocity divided by the change in time. It's also a vector quantity.

5. Instantaneous Acceleration:

• a = dv/dt = d²x/dt²: Instantaneous acceleration is the acceleration at a specific instant, the derivative of velocity with respect to time (or the second derivative of position).

6. Uniformly Accelerated Motion (constant acceleration):

These equations are only valid when acceleration is constant:

• v_f = v_i + at: Final velocity equals initial velocity plus acceleration multiplied by time.
• Δx = v_it + (1/2)at²: Displacement equals initial velocity multiplied by time plus one-half acceleration multiplied by time squared.
• v_f² = v_i² + 2aΔx: The square of the final velocity equals the square of the initial velocity plus twice the acceleration times the displacement.

Example: A car accelerates uniformly from rest (v_i = 0 m/s) to 20 m/s in 5 seconds. Find its acceleration and the distance it travels.

*Using v_f = v_i + at, we get a = (20 m/s - 0 m/s) / 5 s = 4 m/s².

Using Δx = v_it + (1/2)at², we get Δx = 0 + (1/2)(4 m/s²)(5 s)² = 50 m.

II. Forces and Newton's Laws

Newton's laws of motion are fundamental to understanding how forces affect the motion of objects.

1. Newton's First Law (Inertia): An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

2. Newton's Second Law:

• F_net = ma: The net force acting on an object is equal to the mass of the object times its acceleration. This is a vector equation; the direction of the net force determines the direction of the acceleration.

3. Newton's Third Law: For every action, there is an equal and opposite reaction. Forces always come in pairs.

4. Weight:

• W = mg: Weight is the force of gravity acting on an object, equal to the mass of the object times the acceleration due to gravity (g ≈ 9.8 m/s² on Earth).

5. Friction:

• f_k = μ_kN: Kinetic friction (friction when objects are moving) is equal to the coefficient of kinetic friction (μ_k) times the normal force (N).
• f_s ≤ μ_sN: Static friction (friction when objects are at rest) is less than or equal to the coefficient of static friction (μ_s) times the normal force. The maximum static friction is μ_sN.

Example: A 10 kg block is pushed across a horizontal surface with a force of 20 N. If the coefficient of kinetic friction is 0.2, what is the acceleration of the block?

*The normal force is equal to the weight: N = mg = (10 kg)(9.8 m/s²) = 98 N.

*The kinetic friction force is f_k = μ_kN = (0.2)(98 N) = 19.6 N.

*The net force is F_net = 20 N - 19.6 N = 0.4 N.

Using F_net = ma, we get a = F_net/m = 0.4 N / 10 kg = 0.04 m/s².

III. Work, Energy, and Power

This section delves into the concepts of energy and its transformations.

1. Work:

• W = Fdcosθ: Work is done when a force causes a displacement. It's equal to the force multiplied by the displacement multiplied by the cosine of the angle between the force and displacement vectors.

2. Kinetic Energy:

• KE = (1/2)mv²: Kinetic energy is the energy of motion, proportional to the mass and the square of the velocity.

3. Potential Energy (Gravitational):

• PE_g = mgh: Gravitational potential energy is the energy stored due to an object's position in a gravitational field. It's equal to the mass times the acceleration due to gravity times the height.

4. Potential Energy (Elastic):

• PE_elastic = (1/2)kx²: Elastic potential energy is the energy stored in a spring that's stretched or compressed. It's equal to one-half times the spring constant (k) times the square of the displacement (x).

5. Work-Energy Theorem:

• W_net = ΔKE: The net work done on an object is equal to the change in its kinetic energy.

6. Conservation of Mechanical Energy (no non-conservative forces):

• KE_i + PE_i = KE_f + PE_f: In the absence of non-conservative forces (like friction), the total mechanical energy (kinetic plus potential) remains constant.

7. Power:

• P = W/Δt = Fv: Power is the rate at which work is done. It can also be expressed as the force multiplied by the velocity.

Example: A 2 kg ball is dropped from a height of 10 m. Find its speed just before it hits the ground (ignoring air resistance).

*Using conservation of mechanical energy: PE_i = KE_f

*mgh = (1/2)mv²

v = √(2gh) = √(2 * 9.8 m/s² * 10 m) ≈ 14 m/s

IV. Linear Momentum and Impulse

Momentum and impulse are crucial concepts in analyzing collisions and interactions between objects.

1. Linear Momentum:

• p = mv: Linear momentum is the product of an object's mass and its velocity. It's a vector quantity.

2. Impulse:

• J = Δp = FΔt: Impulse is the change in momentum. It's also equal to the average force multiplied by the time interval over which the force acts.

3. Conservation of Linear Momentum (in a closed system with no external forces):

• m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f: The total momentum before a collision equals the total momentum after the collision.

Example: A 1 kg cart moving at 2 m/s collides with a stationary 2 kg cart. After the collision, the 1 kg cart moves at -1 m/s. What is the velocity of the 2 kg cart after the collision?

*Using conservation of momentum: (1 kg)(2 m/s) + (2 kg)(0 m/s) = (1 kg)(-1 m/s) + (2 kg)v_2f

Solving for v_2f, we get v_2f = 1.5 m/s

V. Rotational Motion

Rotational motion introduces concepts analogous to linear motion, but dealing with rotations instead of translations.

1. Angular Displacement:

• θ: Measured in radians.

2. Angular Velocity:

• ω = Δθ/Δt: Rate of change of angular displacement.

3. Angular Acceleration:

• α = Δω/Δt: Rate of change of angular velocity.

4. Relationship between Linear and Angular Quantities:

• v = rω: Linear velocity is equal to the radius multiplied by the angular velocity.
• a_t = rα: Tangential acceleration is equal to the radius multiplied by the angular acceleration.
• a_c = v²/r = rω²: Centripetal acceleration is equal to the square of the linear velocity divided by the radius (or radius times the square of the angular velocity).

5. Torque:

• τ = rFsinθ: Torque is the rotational equivalent of force. It's equal to the radius multiplied by the force and the sine of the angle between them.

6. Moment of Inertia:

• I: A measure of an object's resistance to changes in its rotational motion. Its value depends on the object's mass distribution and shape.

7. Rotational Kinetic Energy:

• KE_rot = (1/2)Iω²: The kinetic energy associated with rotational motion.

VI. Simple Harmonic Motion (SHM)

Simple harmonic motion describes oscillatory motion where the restoring force is proportional to the displacement.

1. Period (T): The time for one complete cycle.

2. Frequency (f): The number of cycles per unit time (f = 1/T).

3. Angular Frequency (ω): ω = 2πf = 2π/T

4. Displacement in SHM:

• x(t) = Acos(ωt + φ): Displacement as a function of time, where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

5. Velocity in SHM:

• v(t) = -Aωsin(ωt + φ): Velocity as a function of time.

6. Acceleration in SHM:

• a(t) = -Aω²cos(ωt + φ) = -ω²x(t): Acceleration as a function of time. Note that acceleration is proportional to displacement and opposite in direction.

7. Period of a Simple Pendulum:

• T = 2π√(L/g): The period of a simple pendulum depends on its length (L) and the acceleration due to gravity (g).

8. Period of a Mass-Spring System:

• T = 2π√(m/k): The period of a mass-spring system depends on the mass (m) and the spring constant (k).

VII. Waves

Waves transfer energy without transferring matter.

1. Wave Speed:

• v = fλ: Wave speed is equal to the frequency multiplied by the wavelength.

2. Superposition Principle: When two or more waves overlap, the resulting displacement is the sum of the individual displacements.

3. Interference: The superposition of waves can lead to constructive (amplitudes add) or destructive (amplitudes subtract) interference.

4. Diffraction: The bending of waves as they pass through an opening or around an obstacle.

VIII. Electric Circuits

This section covers the basics of electric circuits and their components.

1. Ohm's Law:

• V = IR: The voltage across a resistor is equal to the current through the resistor multiplied by its resistance.

2. Electric Power:

• P = IV = I²R = V²/R: Electric power is the rate at which energy is transferred in a circuit.

3. Series Circuits: Resistors connected end-to-end. The total resistance is the sum of the individual resistances (R_total = R₁ + R₂ + ...). The current is the same throughout the circuit.

4. Parallel Circuits: Resistors connected across each other. The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances (1/R_total = 1/R₁ + 1/R₂ + ...). The voltage is the same across each resistor.

IX. Important Constants and Conversions

• g (acceleration due to gravity): Approximately 9.8 m/s²
• k (Coulomb's constant): Approximately 8.99 x 10⁹ N⋅m²/C²

X. Tips for Using Your AP Physics 1 Formula Sheet

• Understanding, not memorization: Focus on understanding the underlying concepts and the relationships between variables. Rote memorization will not suffice.
• Practice, practice, practice: Work through numerous problems to apply the formulas in different contexts.
• Draw diagrams: Visualizing the problem with a clear diagram can help you identify relevant formulas and variables.
• Check units: Ensure that your units are consistent throughout your calculations.
• Significant figures: Pay attention to significant figures in your answers.

This comprehensive guide provides a solid foundation for your AP Physics 1 preparation. Remember that consistent effort, a deep understanding of the concepts, and ample practice are key to success. Good luck!