Kinematic Equations and Motion Concepts

Jul 21, 2025

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Overview

This lecture covers the final two kinematic equations for linear motion, the distinction between horizontal and vertical motion, the difference between scalars and vectors, and how to solve average speed problems in physics.

Kinematic Equations for Linear Motion

  • The first new kinematic equation: ( x_f = x_i + v_i t + \frac{1}{2} a t^2 ) (finds position using initial position, initial velocity, acceleration, and time).
  • Multiplication is implied when two variables are written next to each other (e.g., ( v_i t )).
  • In these equations, ( t ) means "change in time" (( \Delta T )) if not otherwise specified.
  • SI units: time (s), position (m), velocity (m/s), acceleration (m/sΒ²).
  • The second new kinematic equation: ( v_f^2 = v_i^2 + 2a(x_f - x_i) ) (finds final velocity without using time).
  • Every kinematic equation can be adapted for horizontal (x) or vertical (y) motion by changing the variables/subscripts.

Horizontal and Vertical Motion Variables

  • Use ( x ) for horizontal position, ( y ) for vertical position (height).
  • Subscripts indicate direction: ( v_x ) (horizontal velocity), ( v_y ) (vertical velocity), ( a_x ), ( a_y ).
  • Time (( t )) does not get a directional subscript.

Understanding Variables and Subscripts

  • Variables: ( T ) (time), ( x )/( y ) (position), ( v ) (velocity), ( a ) (acceleration).
  • Subscripts: ( i ) or 0 (initial), ( f ) (final), ( x ) (horizontal), ( y ) (vertical).
  • ( \Delta ) (delta) means "change in," or "final minus initial."

Scalars vs. Vectors

  • Scalar: quantity with only magnitude (e.g., 6 miles, 12 m/s).
  • Vector: quantity with both magnitude and direction (e.g., 6 miles north).
  • Distance is a scalar; displacement is a vector.
  • Speed equals distance over time (scalar); velocity equals displacement over time (vector).

Distance, Displacement, Speed, and Velocity Examples

  • If someone walks a circle of 15 m and returns to the start, distance = 15 m, displacement = 0.
  • Car’s odometer measures distance (scalar), not displacement.
  • Average speed = total distance / total time.
  • Average velocity = total displacement / total time (includes direction).
  • Speedometer shows speed, not velocity.

Average Speed Problems

  • Average speed is calculated using total distance divided by total time, not by averaging speeds.
  • Example: Traveling 150 mi at 50 mph, then 150 mi at 75 mph, average speed = 300 mi / 5 h = 60 mph.
  • Always calculate total time to find average speed, especially if speeds differ for different trip segments.

Key Terms & Definitions

  • Kinematic Equations β€” Equations that describe motion using variables for position, velocity, acceleration, and time.
  • Scalar β€” Quantity with only magnitude, no direction.
  • Vector β€” Quantity with both magnitude and direction.
  • Distance β€” Total path traveled (scalar).
  • Displacement β€” Change in position from start to finish (vector).
  • Speed β€” Rate of distance traveled per unit time (scalar).
  • Velocity β€” Rate of displacement per unit time (vector).
  • Subscript β€” Small letter after a variable indicating initial, final, or direction.
  • Delta (Ξ”) β€” Symbol meaning "change in."

Action Items / Next Steps

  • Review and become familiar with all kinematic equations and their variables/subscripts.
  • Practice applying kinematic equations for both horizontal (x) and vertical (y) motion.
  • Complete homework or practice problems on average speed and distinguishing between scalars and vectors.

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