Week 1: Classical Kinematics

Learning Outcomes

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• At the end of this week, students should be able to:

  • Define physical parameters that describe the behavior of objects in motion.

  • Distinguish between vector and scalar quantities and compute vector analysis using the component method.

  • Identify physical situations in which there is motion with constant acceleration.

  • Interpret velocity-time graphs, position-time graphs, and acceleration-time graphs.

  • Derive kinematics equations from the first definition of kinematic variables.

  • Use equations of motion to solve constant acceleration problems.

  • Evaluate projectile motion problems in 1D and 2D.

1.0 Introduction

• The study of Physics serves fundamental purposes across various fields such as Chemistry, Biology, etc. Physics attempts to understand the nature of things and how they work, making it the purest of sciences.

1.1 Kinematics in One Dimension

• Kinematics is defined as the study of motion without considering its causes. Motion is quantitatively described by the following parameters:

  • Displacement

  • Position

  • Distance

  • Speed

  • Velocity

  • Acceleration

  • Time

1.1.1 Displacement

• Displacement measures the distance from a reference point in space and describes an object's position at a particular instant in time.

• To describe an object's motion:

  • A coordinate system is associated with a reference point, the origin.

  • Displacement (\Delta x) is the vector pointing from the initial position to the final position. It contains both magnitude and direction.

  • Example: If an object starts at position $x_0 = 2.0 ext{ m}$ and moves to $x = 7.0 ext{ m}$, then:

  • Displacement \Delta x = 7.0 ext{ m} - 2.0 ext{ m} = 5.0 ext{ m} (to the right).

  • If it moves from $x = 7.0 ext{ m}$ back to $x = 2.0 ext{ m}$, then:

  • Displacement \Delta x = 2.0 ext{ m} - 7.0 ext{ m} = -5.0 ext{ m} (to the left).

• Note: Do not confuse distance (a scalar quantity with magnitude only) with displacement.

1.1.2 Speed and Velocity

• Average Speed:

  • Defined as distance traveled divided by time required:
    $ext{Average Speed} = rac{ ext{Distance}}{ ext{Time Elapsed}}$

  • SI Unit: $ext{m/s}$ (scalar quantity).

• Average Velocity:

  • Given by the formula:
    $ext{Average Velocity} = rac{ ext{Displacement}}{ ext{Time Elapsed}}$

  • SI Unit: $ext{m/s}$ (vector quantity).

1.1.3 Instantaneous Velocity

• Indicates speed and direction at each instant of time. The average velocity over a long trip does not describe your instantaneous speed.

• Mathematically, defined as the limit:
  $v = rac{ ext{d}x}{ ext{d}t}$
  (as the time interval approaches zero).

1.1.4 Acceleration

• Defined as the rate of change of velocity. Average acceleration can be illustrated using a plane:

  • $ext{Average Acceleration} = rac{ ext{Change in Velocity}}{ ext{Time Elapsed}} = rac{v_f - v_i}{t}$

  • SI Unit: $ext{m/s}^2$ (vector quantity).

• Examples:

  • Example 1: A plane starts from rest and reaches a velocity of +72 m/s at t = 29 s. Average acceleration is:
    $a = rac{v_f - v_i}{t} = rac{72 ext{ m/s} - 0}{29 ext{ s}} ext{m/s}^2 <br>ightarrow 2.5 ext{ m/s}^2$

  • Example 2: A drag racer slows down from 28 m/s to 13 m/s in 3 seconds:

  • Initial velocity $v_0 = 28 ext{ m/s}$, final velocity $v = 13 ext{ m/s}$. The average acceleration becomes:
    $a = rac{13 ext{ m/s} - 28 ext{ m/s}}{3 ext{ s}} = -5 ext{ m/s}^2$
    (the negative sign indicates direction opposite to initial velocity).

1.2 Graphical Analysis of Velocity and Acceleration

• Graphical analysis aids in visualizing motion.

  • Position vs. Time Graph (constant velocity): The slope (rise/run) gives average velocity.

  • Velocity vs. Time Graph (constant acceleration): The slope gives acceleration.

  • Example 3: Analyzing a position-time graph for a bicyclist with segments depicting different velocities.

1.3 Equations of Motion

• For an object moving with constant acceleration, a set of equations of motion applies:

  • Initial velocity $v_i$, final velocity $v_f$, acceleration $a$, time $t$, and displacement $x$ need to be known.

• Equations:

  1. $v_f = v_i + at$

  2. $x = v_i t + rac{1}{2}at^2$

  3. $v_f^2 = v_i^2 + 2ax$

• Example 4: A speedboat accelerates at 4.0 m/s². Given $v_i = 12 ext{ m/s}$, find displacement after 10 seconds:

  • Apply equations: $x_f = v_i t + rac{1}{2}at^2$
    => $x_f = 12(10) + rac{1}{2}(4)(10^2) = 120 + 200 = 320 ext{ m}$. \

  • Find final velocity using $v_f = v_i + at$
    => $v_f = 12 + 4(10) = 52 ext{ m/s}$.

• Example 5: A jogger accelerates from rest to 3 m/s in 2 s:

  • Calculate acceleration and distance covered using the equations of motion.

1.3 Free Falling Bodies

• In free-fall motion, neglecting air resistance, objects fall with the same acceleration due to gravity $g <br>eq 9.8 ext{ m/s}^2$.

• Example 7: A stone drops from rest at the Reserve Bank Building:

  • After 3 s:

  • Displacement $y = rac{1}{2}gt^2 = rac{1}{2}(9.8)(3^2) = 44.1 ext{ m}$

• Example 8: A referee tosses a coin upward:

  • The time in air is twice the time to reach maximum height.

1.4 Kinematics in 2-Dimensions

• Motion involves independent x and y components. Two sets of kinematic equations are needed.

  • Analyze 2D motion by treating two 1D motions separately.

  • Figures depict spacecraft motion with different accelerations along x and y axes respectively.

1.4.1 Vector Analysis

• Vectors represented by arrows showing direction and magnitude. Component calculations utilize trigonometry:

  • Magnitude of x component: $A_x = A ext{cos}( heta)$

  • Magnitude of y component: $A_y = A ext{sin}( heta)$

1.4.2 Vector Addition by Components

• Vectors can be summed by their components:

  • If $R = A + B$, then:

  • $R_x = A_x + B_x$

  • $R_y = A_y + B_y$

1.5 Projectile Motion

• A type of 2D motion with only gravitational acceleration acting vertically.

• The horizontal component of velocity is constant. Key equations govern projectile motion and its trajectory.

  • Example 10: Football kicked with initial velocity at an angle, resolving into components to calculate maximum height and range.

1.6 Conclusion

• Physics is quantitative; it expresses relationships in mathematical forms.

  • The SI units for length, mass, and time are meter, kilogram, and second respectively.

• Kinematics addresses motion, subdivided into straight-line (rectilinear) motion allowing for use of scalar quantities, eventually relating positions, velocities, and accelerations mathematically.

Self-Assessment Problems

• Students are encouraged to work on self-assessment problems to assess understanding.

• Tutorial assignment questions are required to be submitted online. Documents are to be clearly scanned or typed with student identification included.

References

• Cutnell, J. D., & Johnson, K. W. (2012). Physics (9th ed.). John Wiley & Sons, Inc. Australia.