Kinematics Equations

• Kinematics is the study of motion, focusing on displacement, velocity, and acceleration without considering the forces that cause the motion.

Kinematics Equations

• These equations describe the motion of objects with constant acceleration in a straight line.

• The equations relate displacement (Δx), initial velocity (vi), final velocity (vf), acceleration (a), and time (t).

• Equation 1: vf = vi + at

  • This equation relates final velocity to initial velocity, acceleration, and time.
  • It's used when displacement is not known or not needed.

• Equation 2: Δx = vi*t + (1/2)at^2

  • This equation relates displacement to initial velocity, time, and acceleration.
  • It's used when final velocity is not known or not needed.

• Equation 3: vf^2 = vi^2 + 2aΔx

  • This equation relates final velocity to initial velocity, acceleration, and displacement.
  • It's used when time is not known or not needed.

• Equation 4: Δx = ((vi + vf)/2)*t

  • This equation relates displacement to initial velocity, final velocity, and time.
  • It's used when acceleration is constant but not explicitly given.

• Problem-solving strategy involves identifying known variables, identifying the unknown variable, choosing the appropriate equation, and solving for the unknown.

• It may be necessary to combine multiple equations to solve a problem if insufficient information is initially provided.

Kinematics Graphs

• Kinematics graphs are visual representations of motion, including position vs. time, velocity vs. time, and acceleration vs. time.

• Position vs. Time Graphs

  • The slope of a position vs. time graph represents the object's velocity.
  • A straight line indicates constant velocity.
  • A curved line indicates changing velocity (acceleration).
  • The steeper the slope, the greater the velocity.
  • A horizontal line indicates the object is at rest.

• Velocity vs. Time Graphs

  • The slope of a velocity vs. time graph represents the object's acceleration.
  • A straight line indicates constant acceleration.
  • A horizontal line indicates constant velocity (zero acceleration).
  • The area under a velocity vs. time graph represents the object's displacement.
  • An increasing slope indicates positive acceleration.
  • A decreasing slope indicates negative acceleration (deceleration).

• Acceleration vs. Time Graphs

  • The area under an acceleration vs. time graph represents the change in velocity.
  • A horizontal line at zero indicates constant velocity.
  • A horizontal line above zero indicates constant positive acceleration.
  • A horizontal line below zero indicates constant negative acceleration.

• Relationships between Graphs

  • Velocity is the derivative of position with respect to time.
  • Acceleration is the derivative of velocity with respect to time (the second derivative of position with respect to time).
  • Integration can be used to find velocity from acceleration and position from velocity.

Projectiles

• Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity and air resistance (air resistance is often ignored for simplicity).

• Projectile motion is analyzed by considering the horizontal and vertical components of motion separately.

• Horizontal Motion

  • In the absence of air resistance, there is no horizontal acceleration.
  • Horizontal velocity (vx) is constant throughout the motion.
  • Δx = vxt, where vx = vicos(θ) and θ is the launch angle.

• Vertical Motion

  • The only acceleration is due to gravity (g ≈ 9.8 m/s²), acting downward.
  • The vertical velocity (vy) changes due to gravity.
  • Kinematics equations apply to the vertical component of motion:
    • vf_y = vi_y + a*t, where a = -g
    • Δy = vi_y*t + (1/2)at^2, where a = -g
    • vf_y^2 = vi_y^2 + 2aΔy, where a = -g
  • The initial vertical velocity is vi_y = vi*sin(θ).

• Key Concepts in Projectile Motion

  • Time of flight: The total time the projectile is in the air; determined by the vertical motion.
  • Maximum height: The highest vertical position reached by the projectile; occurs when vertical velocity is zero.
  • Range: The horizontal distance traveled by the projectile; depends on initial velocity and launch angle.
  • Symmetry: For level ground, the time to reach maximum height is half the total time of flight, and the launch angle and landing angle are equal.

• Problem-Solving Strategy for Projectiles

  • Resolve initial velocity into horizontal and vertical components.
  • Analyze vertical motion to find time of flight and/or maximum height.
  • Analyze horizontal motion to find range.
  • Remember that the time is the same for both horizontal and vertical motion.

• Launch Angle and Range

  • For a given initial speed, the maximum range is achieved when the launch angle is 45 degrees (assuming level ground and negligible air resistance).
  • Complementary angles (e.g., 30 degrees and 60 degrees) will result in the same range (assuming level ground and negligible air resistance).

• Effects of Air Resistance

  • Air resistance reduces the range and maximum height of a projectile.
  • Air resistance causes the trajectory to be non-parabolic.
  • Air resistance depends on the projectile's shape, size, and speed.

• Projectiles Launched at an Angle on Inclined Planes

  • The range equation needs to be adjusted to account for the angle of the inclined plane.
  • The time of flight and maximum height calculations also need to consider the inclined plane angle.

• Relative Velocity in Projectile Motion

  • The velocity of a projectile can be measured relative to different frames of reference.
  • Vector addition is used to find the projectile's velocity relative to a stationary observer.