Physics for Computing - Uniform Accelerated Motion Study Notes

COMMUNICATION TECHNOLOGY GCTU UNIVERSITY

  • Course: Physics for Computing (CSPS 132)

  • Topic: Uniform Accelerated Motion (Kinematics)

  • Presented by: Dr. Mark Ofori Nketia, Lecturer

Overview of Accelerated Motion

  • Objectives of Chapter 3:

    • Develop descriptions of accelerated motions.

    • Use graphs and equations to solve problems involving moving objects.

    • Describe the motion of objects in free fall.

Types of Velocity

Instantaneous Velocity

  • Definition: The speed at which an object is moving at a specific moment in time.

Average Velocity

  • Definition: The average speed of an object over a specified time interval.

  • Examples:

    1. Speedometer indicating 65 mph reflects instantaneous velocity.

    2. A race-car driver maintaining 120 mph for the entire race is an example of average velocity.

    3. A freely falling object at 19.6 m/s after 2 seconds of fall is instantaneous velocity.

    4. The speed limit sign (45 mph) also indicates average velocity.

Acceleration

Changing Velocity

  • Indicators of Change in Velocity:

    • The spacing between points in a motion diagram.

    • Length differences of the velocity vectors representing velocity changes.

Definitions of Acceleration

  • Acceleration: The rate at which an object's velocity changes.

  • Constant Acceleration: When velocity changes at a constant rate.

  • Average Acceleration (a̅):

    • Formula:
      a̅ = rac{Δv}{Δt}

    • Units: measured in meters per second squared (m/s²).

  • Instantaneous Acceleration:

    • Found by drawing a tangent line on a velocity-time graph at a given moment; the slope of the line equals the instantaneous acceleration.

Approach to Finding Acceleration

  1. Average Acceleration Calculation:

    • Plot two points on a graph representing a time interval and find the slope of a line connecting the points.

  2. Instantaneous Acceleration Example:

    • Tangent lines at specific time points illustrate changes in acceleration.

Analyzing Graphs

Velocity-Time Graph Analysis

  • Sprinter's Motion Description:

    • Initial velocity at zero, rapid increase, stabilizing around 10.0 m/s.

  • Finding Instantaneous Accelerations:

    • Draw tangents at specified time intervals (e.g., t=1s, t=5s) and calculate slope.

Average Acceleration Example

  • Calculation: Draw a connecting line between time points for average acceleration. Example:

    • Slope between t=1s and t=5s gives average acceleration = 0.8 m/s²:

    • ext{Average Acceleration} = rac{ ext{slope}}{5 ext{s}} = 0.8 ext{ m/s}².

Direction of Velocity and Acceleration

  • Relationship of Vectors:

    • When speeding up, velocity and acceleration vectors point in the same direction (cases 1 & 3).

    • When slowing down, vectors point in opposite directions (cases 2 & 4).

  • Signs of Acceleration:

    • Positive if direction of acceleration is positive; negative if opposite.

    • The sign does not indicate whether the object is speeding up or slowing down.

Velocity-Time Graph Interpretation

  • Positive velocity indicates motion in a designated direction (e.g., east).

  • Zero slope indicates constant velocity, while varying slopes show acceleration changes.

Equations of Motion for Constant Acceleration

  1. First Equation of Motion: v = u + at

    • Relates final velocity (v) to initial velocity (u), acceleration (a), and time (t).

  2. Second Equation of Motion: s = ut + rac{1}{2}at^{2}

    • Relates displacement (s) to initial velocity, acceleration, and time.

  3. Third Equation of Motion: v^2 = u^2 + 2as

    • Relates final velocity to initial velocity, acceleration, and displacement.

  4. Fourth Equation of Motion: s = vt - rac{1}{2}at^{2}

    • Relates displacement to final velocity, acceleration, and time.

Understanding Position-Time and Velocity-Time Graphs

  • Displacement is indicated by the area under the velocity-time graph.

Free Fall

Definition and Characteristics

  • Free Fall: Motion of a body when air resistance is negligible, and gravity is the only acting force.

  • Acceleration due to Gravity (g):

    • Value: g = 9.80 ext{ m/s}^2

    • Galileo's Conclusion: All objects in free fall experience the same acceleration, regardless of mass.

Application to Real Life

  • Concept employed in amusement park rides simulating free fall, demonstrating that different masses experience the same acceleration.

Example of Free Fall Calculation

  • If an object starts at rest and falls for 1.5 seconds:

    • Initial velocity (vi) = 0.0 m/s, acceleration (a) = -9.8 m/s².

    • Calculate final velocity (vf) and displacement (d) using equations for constant acceleration.

Free Fall Q&A

  1. What is Free Fall?

    • Motion due solely to gravity, neglecting air resistance.

  2. Example Problem:

    • If a stone is thrown up at 25 m/s, its velocity after 1s using gravity's downward pull will be lower due to opposite direction of acceleration.
      A: 15.2 m/s.

  3. Comparison of Objects:

    • Different masses dropped simultaneously will fall with the same acceleration due to gravitational pull.