Kinematics: Motion in One Dimension

Chapter 02: Kinematics

Overview of Kinematics

• Description of kinematics as the study of motion without accounting for the forces that cause this motion.

• Fundamental concepts include position, velocity, speed, and acceleration.

Key Concepts in Kinematics

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• Position x: The location of a particle with respect to a chosen reference point.

  • Example from the text:

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  • A car is moving to the right with respect to reference points A and B.

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  • Positions at various times:
    

    Time (s)	Position (m)
    0	30
    10	52
    20	38
    30	0
    40	-37
    50	-53
    Displacement riangle x: Change in position over a time interval, defined mathematically as:
    riangle x = xf - xi
    Where:
    x_f: final position
    x_i: initial position
    Distance vs. Displacement:
    Distance: A scalar quantity that denotes the total length of the path traveled irrespective of direction.
    Displacement: A vector quantity that describes the shortest path between the initial and final position and requires both magnitude and direction.
    Velocity and Speed

    • Average velocity v_{avg}: Defined as the displacement divided by the time interval:

      • v_{avg} = rac{ riangle x}{ riangle t}

      • Example calculation: A car moves from position 30 m to 52 m in 10 s:
        v_{avg} = rac{52 m - 30 m}{10 s - 0 s} = 2.2 m/s

    • Average speed: A scalar quantity given by the total distance traveled divided by the total elapsed time:

      • Example: If total distance is 75 m over 55 s, then:
        V_{avg} = rac{75 m}{55 s} = 1.36 m/s

    Instantaneous Velocity

    • Instantaneous velocity v: The velocity of an object at a specific moment in time, represented mathematically as:

      • v = rac{dx}{dt}

      • Obtained by taking the derivative of the position function with respect to time.

    Concepts of Acceleration

    • Acceleration a: The rate of change of velocity over time, defined as:

      • a = rac{ riangle v}{ riangle t}

      • If the acceleration is constant, we can employ kinematic equations to describe motion.

    Motion Diagrams

    • Visual representations that depict the positions of a particle over time, helping to conceptualize its motion.

    • Types of motion diagrams include:

      • Constant velocity: Particles are equidistant over time intervals.

      • Constant acceleration: Distances increase progressively larger in time intervals.

    Kinematic Equations for Constant Acceleration

    1. xf = xi + v_i t + rac{1}{2} a t^2

       • Relates initial and final position with time and acceleration.

    2. vf = vi + at

       • Relates initial and final velocities with acceleration and time.

    3. vf^2 = vi^2 + 2a(xf - xi)

       • Relates velocities with position and acceleration.

    Example Problems

    Example 2.1: Average Velocity and Speed Calculation

    • A detailed calculation from the car's motion:

      • From position A (30 m) to position F (-53 m):

      • Displacement: riangle x = -53 - 30 = -83 m

      • Average velocity:
        v_{avg} = rac{-83 m}{50 s} = -1.66 m/s

      • Distance traveled: 22 m from A to B and 105 m from B to F, total 127 m.

      • Average speed:
        V_{avg} = rac{127 m}{50 s} = 2.54 m/s

    Freely Falling Objects

    • Free Fall: Objects accelerate downward under gravity, undergoing constant acceleration throughout the fall:

      • Acceleration due to gravity is g = 9.8 m/s^2.

      • The kinematic equations apply with downward direction defined as positive for consistent sign use.

    • Example: A stone thrown up with initial velocity will decelerate until it reaches the maximum height (velocity = 0) and then accelerate downward.

    Analysis Models

    • Particle under constant velocity: Simplifies problems by keeping velocity fixed.

      • Example equations for displacement using constant velocities.

    • Particle under constant acceleration: Enables complex motion analysis with consistently changing speeds.

      • Uses derived kinematic formulas to solve for position, velocity, and time.

    Conceptual Questions and Quizzes

    • Quick quizzes included at various points prompt critical thinking about the application of concepts.

    • For example:

      • Question: Under what conditions is the magnitude of average velocity less than average speed?

    Conclusion

    • Understanding these fundamental kinematics concepts lays the groundwork for further studies in physics, motion dynamics, and force applications. Kinematics is vital for any study of mechanics, from analyzing simple movements to complex dynamic systems.