⚙️AP Physics C: Mechanics (2025) Unit 1 – Kinematics

Key Concepts and Definitions

• Kinematics involves the study of motion without considering the forces causing it
• Displacement measures the shortest distance and direction between two points (vector quantity)
• Velocity describes the rate of change of an object's position in a particular direction (vector quantity)
  • Average velocity calculated by dividing the displacement by the time interval ($v_{avg} = \frac{\Delta x}{\Delta t}$)
  • Instantaneous velocity represents the velocity at a specific instant in time (limit of average velocity as time interval approaches zero)
• Acceleration measures the rate of change of velocity (vector quantity)
  • Average acceleration calculated by dividing the change in velocity by the time interval ($a_{avg} = \frac{\Delta v}{\Delta t}$)
  • Instantaneous acceleration represents the acceleration at a specific instant in time (limit of average acceleration as time interval approaches zero)
• Scalar quantities have magnitude only (speed, distance, time)
• Vector quantities have both magnitude and direction (displacement, velocity, acceleration)

Motion in One Dimension

• One-dimensional motion occurs along a straight line
• Position, velocity, and acceleration are described using scalar quantities (positive or negative values indicate direction)
• Kinematic equations for constant acceleration:
  • $v = v_0 + at$
  • $x = x_0 + v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2a(x - x_0)$
• Objects under constant acceleration exhibit a linear velocity-time graph and a parabolic position-time graph
• Freely falling objects experience constant acceleration due to gravity (9.8 m/s²) in the absence of air resistance
• Kinematic equations can be used to solve problems involving displacement, velocity, acceleration, and time

Vectors and Two-Dimensional Motion

• Two-dimensional motion involves movement in both the x and y directions
• Vectors represent quantities with both magnitude and direction (displacement, velocity, acceleration)
• Vector addition follows the parallelogram law or the head-to-tail method
  • Resultant vector found by adding the x and y components separately
• Vector subtraction involves adding the negative of the vector being subtracted
• Scalar multiplication changes the magnitude of a vector without altering its direction
• Unit vectors (i, j) represent the directions along the x and y axes, respectively
• Dot product of two vectors yields a scalar quantity (A · B = AB cos θ)
• Cross product of two vectors yields a vector quantity perpendicular to both input vectors (A × B = AB sin θ n)

Projectile Motion

• Projectile motion combines horizontal and vertical motion of an object
• Horizontal motion has constant velocity (no acceleration)
• Vertical motion has constant acceleration due to gravity
• Projectile motion problems can be solved by analyzing horizontal and vertical components separately
  • Horizontal displacement: $x = v_0 \cos \theta t$
  • Vertical displacement: $y = v_0 \sin \theta t - \frac{1}{2}gt^2$
• Time of flight depends on the initial velocity and launch angle ($t = \frac{2v_0 \sin \theta}{g}$)
• Range of a projectile is the horizontal distance traveled before hitting the ground ($R = \frac{v_0^2 \sin 2\theta}{g}$)
  • Maximum range achieved at a launch angle of 45° (neglecting air resistance)
• Projectile motion examples include thrown balls, launched rockets, and jumping animals

Relative Motion and Reference Frames

• Relative motion describes the motion of an object with respect to a chosen reference frame
• Reference frames can be stationary or moving with constant velocity
• Velocity addition formula: $v_{AB} = v_{A} - v_{B}$
  • $v_{AB}$: velocity of A relative to B
  • $v_{A}$: velocity of A relative to a stationary reference frame
  • $v_{B}$: velocity of B relative to the same stationary reference frame
• Galilean relativity states that the laws of motion are the same in all inertial reference frames
• Non-inertial reference frames experience fictitious forces (centrifugal force on a merry-go-round)
• Relative motion examples include a passenger walking on a moving train or a boat crossing a river with current

Equations and Problem-Solving Strategies

• Identify the known and unknown quantities in the problem
• Choose the appropriate kinematic equation(s) based on the given information
• Substitute known values into the equation(s) and solve for the unknown quantity
• Check the units and the reasonableness of the answer
• Common kinematic equations for constant acceleration:
  • $v = v_0 + at$
  • $x = x_0 + v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2a(x - x_0)$
• For projectile motion, use separate equations for horizontal and vertical components
• When dealing with relative motion, choose a convenient reference frame and apply the velocity addition formula
• Draw diagrams to visualize the problem and identify the relevant quantities (free-body diagrams, vector diagrams)

Real-World Applications

• Kinematics principles are used in various fields, such as engineering, sports, and transportation
• Analyzing the motion of vehicles (cars, trains, airplanes) for safety and efficiency
• Designing roller coasters and amusement park rides
• Studying the motion of athletes (runners, jumpers, throwers) to optimize performance
  • Usain Bolt's world record 100-meter dash (average speed of 10.44 m/s)
• Investigating car accidents and reconstructing the events based on skid marks and impact damage
• Predicting the trajectory of projectiles in military applications (missiles, bullets)
• Understanding the motion of celestial bodies in astronomy (planets, moons, asteroids)
• Developing video games and computer simulations that incorporate realistic motion

Common Misconceptions and Pitfalls

• Confusing scalar and vector quantities (speed vs. velocity, distance vs. displacement)
• Neglecting to consider the sign of velocity and acceleration when dealing with one-dimensional motion
• Forgetting to separate horizontal and vertical components in projectile motion problems
• Misinterpreting graphs (slope of position-time graph represents velocity, not acceleration)
• Applying kinematic equations to situations with non-constant acceleration
• Incorrectly assuming that an object with zero velocity must have zero acceleration
• Mixing up the order of vector subtraction ($v_{AB} \neq v_{BA}$)
• Attempting to use kinematic equations in non-inertial reference frames without accounting for fictitious forces
• Ignoring the effects of air resistance, which can significantly impact the motion of objects in real-world scenarios