College Physics I – Introduction Unit 2 Review: Kinematics

Key Concepts and Definitions

• Kinematics studies motion without considering the forces causing it
• Displacement ($\Delta x$) represents change in position, a vector quantity measured in meters (m)
• Distance traveled measures total path length, a scalar quantity also measured in meters (m)
• Speed is the rate at which an object covers distance, measured in meters per second (m/s)
  • Instantaneous speed is speed at a specific moment in time
  • Average speed is total distance traveled divided by total time elapsed
• Velocity ($\vec{v}$) is the rate of change of displacement, a vector quantity measured in meters per second (m/s)
  • Instantaneous velocity is velocity at a specific instant in time
  • Average velocity equals displacement divided by time interval ($\bar{v} = \frac{\Delta x}{\Delta t}$)
• Acceleration ($\vec{a}$) is the rate of change of velocity, a vector quantity measured in meters per second squared (m/s²)
  • Positive acceleration occurs when an object speeds up or changes direction in the positive direction
  • Negative acceleration, or deceleration, occurs when an object slows down or changes direction in the negative direction

Motion in One Dimension

• One-dimensional motion occurs along a straight line, either horizontally (x-axis) or vertically (y-axis)
• Position-time graphs show an object's position relative to the origin at various times
  • Slope of the tangent line at any point represents the object's instantaneous velocity
  • Slope of the secant line between two points represents the object's average velocity over that time interval
• Velocity-time graphs display an object's velocity over time
  • Slope of the tangent line at any point represents the object's instantaneous acceleration
  • Area under the curve over a time interval equals the object's displacement during that interval
• Kinematic equations describe motion in terms of displacement ($\Delta x$), initial velocity ($v_0$), final velocity ($v$), acceleration ($a$), and time ($t$):
  • $v = v_0 + at$
  • $\Delta x = v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2a\Delta x$
• Objects under constant acceleration exhibit specific characteristics in their position-time and velocity-time graphs
  • Position-time graph shows a parabolic curve
  • Velocity-time graph appears as a straight line with slope equal to the acceleration

Vectors and Two-Dimensional Motion

• Vectors possess both magnitude and direction, represented by an arrow
  • Magnitude is the length of the arrow, denoting the quantity's size
  • Direction is indicated by the arrow's orientation
• Scalar quantities have magnitude but no direction (distance, speed, time)
• Vector addition follows the head-to-tail method or parallelogram rule
  • Head-to-tail method involves placing the tail of one vector at the head of the other, then drawing a resultant vector from the first vector's tail to the second vector's head
  • Parallelogram rule involves placing the two vectors tail-to-tail, then completing a parallelogram and drawing the resultant vector along the diagonal from the common tail to the opposite corner
• Vector subtraction is achieved by adding the negative of the vector being subtracted ($\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$)
• Two-dimensional motion can be analyzed by breaking vectors into perpendicular components (x and y)
  • Components are found using trigonometric functions (sine and cosine)
  • Motion in each dimension is treated independently, then combined to determine the object's overall motion

Acceleration and Free Fall

• Free fall is motion under the sole influence of gravity, with an acceleration of approximately -9.8 m/s² (denoted as $-g$)
  • Negative sign indicates downward direction
  • Air resistance is assumed to be negligible in most introductory physics problems
• Kinematic equations for free fall are similar to those for constant acceleration, with $a = -g$:
  • $v = v_0 - gt$
  • $\Delta y = v_0t - \frac{1}{2}gt^2$
  • $v^2 = v_0^2 - 2g\Delta y$
• Objects in free fall experience zero velocity at their maximum height
  • Time to reach maximum height can be found by setting $v = 0$ and solving for $t$
  • Maximum height is then determined by substituting this time into the position equation
• Total time of flight for a freely falling object launched upward is twice the time to reach its maximum height
• Acceleration due to gravity is independent of an object's mass or shape, as demonstrated by Galileo's famous Leaning Tower of Pisa experiment

Projectile Motion

• Projectile motion is a combination of horizontal and vertical motion, with gravity acting only in the vertical direction
  • Horizontal velocity remains constant (assuming negligible air resistance)
  • Vertical motion is treated as free fall with an initial velocity component
• To analyze projectile motion, the initial velocity ($v_0$) is resolved into horizontal ($v_{0x}$) and vertical ($v_{0y}$) components using trigonometry
  • $v_{0x} = v_0 \cos \theta$
  • $v_{0y} = v_0 \sin \theta$, where $\theta$ is the launch angle relative to the horizontal
• Time of flight is determined by the vertical motion, setting $\Delta y = 0$ (for landing at the same height as launch) and solving the quadratic equation for $t$
• Range is the horizontal distance traveled by the projectile, found by multiplying the horizontal velocity by the time of flight
  • Maximum range for a given initial speed occurs at a launch angle of 45° (neglecting air resistance)
• Trajectory of a projectile is a parabola, with the shape determined by the launch angle and initial speed

Relative Motion and Frame of Reference

• Motion is always described relative to a chosen frame of reference
  • A frame of reference is a set of coordinates used to specify positions and velocities
  • Common frames of reference include the ground, a moving vehicle, or a coordinate system attached to an object
• Relative velocity is the velocity of an object as observed from a particular frame of reference
  • Relative velocity between two objects is the vector difference of their individual velocities
  • $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$, where $\vec{v}_{AB}$ is the velocity of object A relative to object B
• Galilean velocity transformation relates velocities in different frames of reference
  • $\vec{v} = \vec{v}' + \vec{u}$, where $\vec{v}$ is the velocity in the original frame, $\vec{v}'$ is the velocity in the new frame, and $\vec{u}$ is the velocity of the new frame relative to the original frame
• Relative motion problems often involve objects moving in different directions or frames of reference moving relative to each other (boats in a river current, planes in the presence of wind)

Equations and Problem-Solving Strategies

• Kinematic equations for constant acceleration:
  • $v = v_0 + at$
  • $\Delta x = v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2a\Delta x$
• Equations for free fall (with $a = -g$):
  • $v = v_0 - gt$
  • $\Delta y = v_0t - \frac{1}{2}gt^2$
  • $v^2 = v_0^2 - 2g\Delta y$
• Projectile motion equations:
  • Horizontal motion: $\Delta x = v_{0x}t = (v_0 \cos \theta)t$
  • Vertical motion: $\Delta y = v_{0y}t - \frac{1}{2}gt^2 = (v_0 \sin \theta)t - \frac{1}{2}gt^2$
• Problem-solving strategies:
  1. Identify given information and unknowns, listing variables and values
  2. Determine the appropriate equation(s) for the situation
  3. Solve the equation(s) for the unknown variable(s)
  4. Substitute known values and calculate the result
  5. Check the answer for reasonableness and proper units
• When dealing with vectors, break them into components and treat each dimension separately
• For relative motion problems, clearly define the frames of reference and use Galilean velocity transformation to relate velocities between frames

Real-World Applications and Examples

• Sports:
  • Analyzing the motion of a thrown baseball or kicked soccer ball (projectile motion)
  • Determining the optimal angle for a basketball shot or ski jump (projectile motion)
  • Calculating the relative velocities of players on a field or court (relative motion)
• Transportation:
  • Determining the time for a car to reach a certain speed or distance (constant acceleration)
  • Analyzing the motion of an airplane in the presence of wind (relative motion)
  • Calculating the time for a skydiver to reach the ground (free fall with air resistance)
• Engineering and design:
  • Designing roller coasters with appropriate accelerations and velocities for safety and thrill (constant acceleration, free fall)
  • Analyzing the motion of objects on conveyor belts or assembly lines (relative motion)
  • Determining the trajectory of a water fountain or fireworks display (projectile motion)
• Physics experiments:
  • Demonstrating the independence of horizontal and vertical motion in projectile motion (projectile motion)
  • Measuring the acceleration due to gravity using a free fall apparatus (free fall)
  • Investigating the relationship between position, velocity, and acceleration using motion sensors and graphical analysis (constant acceleration)