Key Equations for Kinematics

Kinematics is all about understanding motion. These key equations help describe how objects move, connecting position, velocity, and acceleration. Mastering these concepts is crucial for tackling problems in AP Physics C: Mechanics and grasping the fundamentals of motion.

1. x = x₀ + v₀t + ½at²

   • Describes the position of an object at time t, given its initial position (x₀), initial velocity (v₀), and constant acceleration (a).
   • Useful for calculating displacement when acceleration is constant.
   • The term ½at² accounts for the additional distance covered due to acceleration.

2. v = v₀ + at

   • Relates final velocity (v) to initial velocity (v₀), acceleration (a), and time (t).
   • Indicates how velocity changes over time under constant acceleration.
   • Essential for determining the final speed of an object after a certain time interval.

3. v² = v₀² + 2a(x - x₀)

   • Connects the squares of the velocities to displacement and acceleration.
   • Useful for finding the final velocity without needing time.
   • Highlights the relationship between kinetic energy and work done by acceleration.

4. x = x₀ + ½(v + v₀)t

   • Averages the initial and final velocities to find displacement over time.
   • Useful when the final velocity is known, or when acceleration is not constant.
   • Provides a way to calculate distance traveled when velocity changes linearly.

5. ⟨v⟩ = (x - x₀) / t

   • Defines average velocity (⟨v⟩) as the total displacement divided by the total time.
   • Important for understanding motion over a time interval.
   • Helps in comparing different segments of motion.

6. ⟨a⟩ = (v - v₀) / t

   • Defines average acceleration (⟨a⟩) as the change in velocity over the time interval.
   • Useful for analyzing how quickly an object speeds up or slows down.
   • Important for understanding the effects of forces acting on an object.

7. r = r₀ + v₀t + ½at²

   • Similar to the first equation but used in vector form for position in two or three dimensions.
   • Accounts for initial position (r₀) and motion in a specific direction.
   • Essential for analyzing motion in a coordinate system.

8. v = dr/dt

   • Defines velocity (v) as the derivative of position (r) with respect to time (t).
   • Fundamental concept in calculus-based physics for understanding motion.
   • Indicates how position changes instantaneously with time.

9. a = dv/dt = d²r/dt²

   • Defines acceleration (a) as the derivative of velocity (v) with respect to time (t) or the second derivative of position (r).
   • Essential for understanding how forces affect motion.
   • Connects kinematics with dynamics through Newton's laws.

10. v = ωr (for circular motion)

    • Relates linear velocity (v) to angular velocity (ω) and radius (r) in circular motion.
    • Important for analyzing objects moving in circular paths.
    • Highlights the relationship between linear and rotational motion.