⚙️AP Physics C: Mechanics Unit 1 – Kinematics

Key Concepts and Definitions

• Kinematics the study of motion without considering the forces causing it
• Displacement the change in position of an object, a vector quantity denoted by $\Delta x$ or $\Delta \vec{r}$
• Velocity the rate of change of position with respect to time, a vector quantity denoted by $\vec{v}$
  • Average velocity calculated as $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$
  • Instantaneous velocity the velocity at a specific instant in time, found by taking the limit of average velocity as $\Delta t$ approaches zero
• Acceleration the rate of change of velocity with respect to time, a vector quantity denoted by $\vec{a}$
  • Average acceleration calculated as $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$
  • Instantaneous acceleration the acceleration at a specific instant in time, found by taking the limit of average acceleration as $\Delta t$ approaches zero
• Scalar quantities have magnitude only (speed, distance)
• Vector quantities have both magnitude and direction (velocity, acceleration, displacement)

Motion in One Dimension

• One-dimensional motion occurs along a straight line, either horizontally (x-axis) or vertically (y-axis)
• Position, velocity, and acceleration are functions of time in one dimension
  • Position function $x(t)$ describes an object's position at any given time
  • Velocity function $v(t)$ describes an object's velocity at any given time
  • Acceleration function $a(t)$ describes an object's acceleration at any given time
• Constant velocity occurs when an object's velocity does not change over time ($a = 0$)
• Constant acceleration occurs when an object's acceleration does not change over time
  • Examples include free fall under gravity and motion on an inclined plane
• Kinematic equations for constant acceleration in one dimension:
  • $v = v_0 + at$
  • $x = x_0 + v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2a(x - x_0)$
• Free fall is a special case of constant acceleration with $a = -g = -9.8 \text{ m/s}^2$ (downward)

Vectors and Two-Dimensional Motion

• Two-dimensional motion occurs in a plane (x-y plane) and requires vector analysis
• Vector components break a vector into its x and y components using trigonometry
  • $\vec{A} = A_x \hat{i} + A_y \hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in the x and y directions
  • $A_x = A \cos \theta$ and $A_y = A \sin \theta$, where $\theta$ is the angle between the vector and the positive x-axis
• Vector addition and subtraction follow the rules of vector algebra
  • Resultant vector $\vec{R} = \vec{A} + \vec{B}$ is found using the parallelogram law or by adding components
  • $R_x = A_x + B_x$ and $R_y = A_y + B_y$
• Relative velocity is the velocity of one object with respect to another
  • $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$, where $\vec{v}_{AB}$ is the velocity of A relative to B
• Projectile motion is a special case of two-dimensional motion with constant acceleration due to gravity

Equations and Graphs

• Kinematic equations relate position, velocity, acceleration, and time
  • $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt}$
  • For constant acceleration: $v = v_0 + at$, $x = x_0 + v_0t + \frac{1}{2}at^2$, and $v^2 = v_0^2 + 2a(x - x_0)$
• Graphs of position, velocity, and acceleration vs. time provide visual representations of motion
  • Slope of position graph represents velocity
  • Slope of velocity graph represents acceleration
  • Area under velocity graph represents displacement
  • Area under acceleration graph represents change in velocity
• Graphs can be used to analyze and interpret motion
  • Constant velocity appears as a straight line on a position vs. time graph
  • Constant acceleration appears as a parabola on a position vs. time graph and a straight line on a velocity vs. time graph
• Equations of motion can be derived from graphs
  • Example: $v = \frac{\Delta x}{\Delta t}$ can be found from the slope of a secant line on a position vs. time graph

Projectile Motion

• Projectile motion is the motion of an object under the influence of gravity alone
• Projectile motion can be analyzed as two independent one-dimensional motions:
  • Horizontal motion with constant velocity (no acceleration)
  • Vertical motion with constant acceleration due to gravity
• Equations for projectile motion:
  • Horizontal: $x = v_0 \cos \theta_0 \cdot t$
  • Vertical: $y = v_0 \sin \theta_0 \cdot t - \frac{1}{2}gt^2$
  • Time of flight: $t = \frac{2v_0 \sin \theta_0}{g}$
  • Range: $R = \frac{v_0^2 \sin 2\theta_0}{g}$
• Projectile motion is symmetrical about its highest point (apex)
  • Time to reach apex equals time from apex to landing
  • Velocity at landing has same magnitude as initial velocity but opposite vertical component
• Projectile problems often involve finding initial velocity, launch angle, time of flight, range, or maximum height

Relative Motion and Reference Frames

• Relative motion is the motion of one object with respect to another
• Reference frames are coordinate systems used to describe motion
  • Inertial reference frames are non-accelerating frames in which Newton's laws hold
  • Non-inertial reference frames are accelerating frames in which fictitious forces appear
• Relative velocity is the velocity of one object as observed from another object's reference frame
  • $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$, where $\vec{v}_{AB}$ is the velocity of A relative to B
• Relative acceleration is the acceleration of one object as observed from another object's reference frame
  • $\vec{a}_{AB} = \vec{a}_A - \vec{a}_B$, where $\vec{a}_{AB}$ is the acceleration of A relative to B
• Galilean transformations relate positions, velocities, and accelerations between inertial reference frames
  • $x' = x - vt$, $v' = v$, and $a' = a$, where primed quantities are in the moving frame and unprimed quantities are in the stationary frame
• Relative motion problems often involve finding relative velocities, relative positions, or relative accelerations between objects or reference frames

Applications and Problem-Solving Strategies

• Identify the knowns and unknowns in the problem
• Draw diagrams to visualize the problem (motion diagrams, free-body diagrams)
• Choose appropriate equations based on the given information and the quantity to be found
• Solve equations symbolically before plugging in numbers
• Check units and significant figures in the final answer
• Analyze the reasonableness of the result based on physical intuition
• Examples of applications:
  • Analyzing the motion of vehicles (cars, trains, planes)
  • Describing the motion of objects in sports (balls, athletes)
  • Investigating the motion of particles in physics experiments
  • Predicting the trajectory of projectiles (bullets, rockets, satellites)
• Problem-solving strategies:
  • Break complex problems into smaller, manageable parts
  • Use symmetry and conservation laws to simplify problems
  • Apply limiting cases (e.g., setting acceleration to zero for constant velocity)
  • Utilize graphs and diagrams to gain insight into the problem
  • Check special cases (e.g., setting time or distance to zero)

Common Misconceptions and FAQs

• Misconception: Velocity is always in the same direction as acceleration
  • Velocity and acceleration are vectors and can point in different directions
  • Example: A car slowing down has velocity and acceleration in opposite directions
• Misconception: An object with zero velocity must have zero acceleration
  • An object can have zero velocity and non-zero acceleration at an instant
  • Example: A ball thrown upward has zero velocity at its peak but non-zero acceleration due to gravity
• Misconception: Heavier objects fall faster than lighter objects
  • In the absence of air resistance, all objects fall with the same acceleration (9.8 m/s²)
  • Example: A feather and a hammer dropped on the Moon will hit the ground at the same time
• FAQ: What is the difference between speed and velocity?
  • Speed is a scalar quantity that describes how fast an object moves, while velocity is a vector quantity that describes both speed and direction
• FAQ: Can an object have constant velocity and non-zero acceleration?
  • No, constant velocity implies zero acceleration, as acceleration is the rate of change of velocity
• FAQ: How do you determine the landing point of a projectile?
  • Set the y-coordinate equal to the landing height (usually zero) and solve for time, then use that time to find the x-coordinate at landing
• FAQ: What is the difference between a reference frame and a coordinate system?
  • A reference frame is a physical system used to describe motion, while a coordinate system is a mathematical tool used to assign positions within a reference frame