Lecture Notes on Velocity and Acceleration

Introduction to One-Dimensional Motion

• Discussion starts with one-dimensional motion of an object.
• Positions of the object at times t1, t2, t3, t4, and t5 defined along a straight line (x).
• Increasing values of x are chosen for convenience.

Average Velocity

• Defined as:
  [ \bar{v} = \frac{x(t_2) - x(t_1)}{t_2 - t_1} ]
• Sign of average velocity:
  • Positive when moving in the increasing direction of x.
  • Zero when returning to the same position (e.g. t1 to t5).
  • Negative if moving in the opposite direction (e.g. t2 to t4).
• Direction of increasing x affects signs:
  • Changing direction flips the signs, but location of zero x is irrelevant.

Distance vs. Displacement

• Average speed defined as:
  [ \text{average speed} = \frac{\text{total distance}}{\text{total time}} ]
• Example of distance calculation from t1 to t5 given positions.
• Average speed could be positive even if average velocity is zero (e.g., 300 m in 3 seconds leads to 100 m/s speed).

Instantaneous Velocity

• Defined using limits:
  [ v = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} ]
• This signifies the first derivative of position with respect to time:
  [ v = \frac{dx}{dt} ]
• Determines whether instantaneous velocity is positive, negative, or zero based on the slope of the position-time graph.

Speed vs. Velocity

• Velocity includes direction:
  • Example: v1 = +30 m/s, v2 = -100 m/s.
  • Speed is the magnitude (e.g., speed = 100 m/s), regardless of direction.

Example of Measuring Bullet Speed

• Setup includes two wires to measure the time it takes for a bullet to travel a distance D.
• Measurement uncertainties need to be calculated for distance and timing.
• Aim for two percent accuracy in speed measurement.
• Example calculation resulted in an average speed of 256 ± 4 m/s.

Introduction to Acceleration

• Average acceleration defined as:
  [ \bar{a} = \frac{v(t_2) - v(t_1)}{t_2 - t_1} ]
• Can be positive, negative, or zero depending on velocity changes.
• Sign conventions for acceleration are similar to velocity.

Instantaneous Acceleration

• Defined as the limit of average acceleration as time interval approaches zero:
  [ a = \lim_{\Delta t \to 0} \frac{v(t + \Delta t) - v(t)}{\Delta t} ]
• Represents the second derivative of position with respect to time:
  [ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} ].

Example Problem: Acceleration and Velocity

• Given:
  [ x = 8 - 6t + t^2 ]
• Velocity derived as:
  [ v = \frac{dx}{dt} = -6 + 2t ]
• Acceleration derived as:
  [ a = \frac{dv}{dt} = 2 ].

Summary of Kinematic Equations for One-Dimensional Motion

• General equations relate position, velocity, and acceleration under constant acceleration:
  • Position:
    [ x = x_0 + v_0 t + \frac{1}{2}a t^2 ]
  • Velocity:
    [ v = v_0 + at ]
  • Constant acceleration a = g for free fall (approximately 9.80 m/s²).

Gravity and Beyond

• Gravitational acceleration is independent of the mass and properties of the falling object (in vacuum).
• Determining gravity through experiments measuring time and distance fallen.

Strobe Light Experiment

• Concept of strobing an apple falling to visualize velocity increase.
• Different frequencies result in varying visibility of the apple's position.

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This lecture covers essential principles of velocity and acceleration in one-dimensional motion. Key definitions and equations were presented with examples for practical understanding of concepts.