Study Notes on Kinematics, Vectors, and Motion in Two Dimensions

Kinematics, Vectors, and Motion in Two Dimensions

• Introduction

  • Welcome to Topic 1.5
  • Daily Video 3: Analyzing the motion of horizontal projectiles.
  • Instructor: Wesley Baker, Creekview High School, Carrollton, Texas.

• Definition of Projectiles

  • Projectiles are defined as objects that move solely under the influence of gravity.
  • They do not have any external force acting on them after they are projected forward; the only force encountered is due to gravity.
  • The only acceleration acting on projectiles is the acceleration due to gravity (denoted as $g$).

• Visual Example of a Horizontal Projectile

  • Description of a steel ball bearing moving across a grid, showing its path.
  • The ball follows a parabolic trajectory, which is demonstrated through the plotting of its position at equal time intervals.
  • This parabolic path indicates both horizontal and vertical motion, which occur independently but simultaneously.

• Vertical Motion Analysis

  • When examining the vertical component of motion:
  • Initial velocity in the vertical direction is zero ($v_{0y} = 0$).
  • The ball experiences free fall, accelerating at $g$ downwards.
  • As it falls, it speeds up due to gravity’s acceleration.
  • Kinematic Equation for Vertical Motion:
  • The displacement in the vertical direction can be expressed as:
    $\Delta Y = v_{0y} imes t + \frac{1}{2} g t^2$
  • Given that $v_{0y} = 0$, this simplifies to:
    $\Delta Y = \frac{1}{2} g t^2$
  • The vertical motion of the projectile is identical to that of an object in free fall, due to having the same initial velocity and acceleration.

• Comparison of Horizontal and Vertical Motion

  • An important point made is that even though a projectile has horizontal velocity, it falls at the same rate as an object in free fall (i.e., dropped from the same height).
  • Horizontal Motion Analysis
  • The horizontal motion is characterized by a constant velocity.
  • As visualized with a set of dots plotted onto the horizontal axis, they illustrate that the distance covered in horizontal motion remains constant over equal time intervals.
  • Kinematic Equation for Horizontal Motion:
  • The position in the horizontal direction can be described by:
    $\Delta X = v<em>{0x} t + \frac{1}{2} a</em>x t^2$
  • Here, since there is no horizontal acceleration ($a<em>x = 0$), the equation simplifies to: $\Delta X = v</em>{0x} imes t$
  • The constant horizontal velocity means that the initial horizontal velocity remains the same throughout the motion.

• Side-by-Side Comparison of Motions

  • In analyzing the two components together:
  • Vertical Motion:
    • Initial velocity: $0 ext{ m/s}$ (not projected vertically).
    • Acceleration: $g$ downwards.
    • Displacement as a function of time: $\Delta Y = \frac{1}{2} g t^2$.
  • Horizontal Motion:
    • Constant velocity: Just the initial horizontal velocity.
    • No acceleration: Horizontal motion does not have vertical influence.
    • Displacement as a function of time: $\Delta X = v_{0x} t$.

• Practical Example/Question

  • Scenario: Two students - one drops a piece of paper straight down, while the other throws an identical piece horizontally from the same height.
  • Question: Which crumbled piece of paper hits the ground first?
  • Answer: Both pieces hit the ground at the same time if air resistance is ignored.
  • The vertical motion (acceleration and height) for both pieces of paper is the same.
  • The only difference is the thrown paper travels a greater horizontal distance due to its horizontal velocity.

• Key Takeaways

  • Vertical and horizontal motion of projectiles are independent.
  • Vertical motion (free fall) involves acceleration due to gravity, leading to downward speed increase.
  • Horizontal motion involves constant velocity with no acceleration, which remains unchanged regardless of vertical falling.

• Conclusion

  • Thank you for watching; remember the distinctions made between vertical and horizontal motions in projectile motion.
  • Emphasize the independence of these two motions and their respective characteristics.