Lecture Notes: Velocities and Acceleration

Introduction to One-Dimensional Motion

• Discussing motion of an object along a straight line.
• Positions of the object at various times:
  • t1: Initial position
  • t2: Position at time t2
  • t3: Position at time t3
  • t4: Position at time t4
  • t5: Back to the position at t1
• Defined increasing value of x.

Average Velocity

• Average velocity defined:
  • ( v_{avg} = \frac{x(t2) - x(t1)}{t2 - t1} )
• Signs of average velocity:
  • Positive between t1 and t2.
  • Zero between t1 and t5.
  • Negative between t2 and t4.
• Direction of increasing x affects the signs.

Instantaneous Velocity

• Instantaneous velocity defined as:
  • ( v = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} )
  • Recognized as the first derivative of position vs. time:
  • ( v = \frac{dx}{dt} )
• Importance of remembering ( v = \frac{dx}{dt} ) for future reference.

Average Speed vs. Average Velocity

• Average speed is defined as:
  • ( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} )
• Example:
  • Between t1 and t5, average speed can differ from average velocity.
  • Example calculation: total distance of 300 m over 3 seconds yields 100 m/s average speed.

Measurement Example: Bullet Speed

• Setup to measure bullet speed:
  • Distance D between two wires.
  • Timer starts when bullet breaks wire I and stops at wire II.
• Measurement accuracy discussed:
  • Distance uncertainty of 0.5 cm.
  • Timing uncertainty of 0.1 ms for accuracy of 2%.
• Example outcome:
  • Measured time: 5.8 ms.
  • Average speed calculated using distance and time, yielding 256 m/s.

Average Acceleration

• Average acceleration defined:
  • ( a_{avg} = \frac{v(t2) - v(t1)}{t2 - t1} )
• Example with a bouncing tennis ball:
  • Velocity at impact and bounces analyzed for acceleration calculation.

Instantaneous Acceleration

• Instantaneous acceleration defined:
  • ( a = \lim_{\Delta t \to 0} \frac{v(t + \Delta t) - v(t)}{\Delta t} )
  • Represents second derivative of position vs. time.
• Important equation:
  • ( a = \frac{dv}{dt} = \frac{d^2x}{dt^2} )

General Equations for Constant Acceleration

• Position as a function of time:
  • ( x = C_1 + C_2 t + C_3 t^2 )
• Derivatives give insight into velocity and acceleration:
  • ( v = C_2 + 2C_3t ) and ( a = 2C_3 )

Gravitational Acceleration

• Gravitational acceleration denoted as ( g ):
  • ( g = 9.80 , \text{m/s}^2 ) in Boston.
• Equations related to free fall:
  • ( x = x_0 + v_0 t + \frac{1}{2} g t^2 )
  • ( v = v_0 + gt )

Experimental Setup

• Strobe light experiment to visualize falling apple:
  • Importance of timing and measurements in experiments.
  • Discussion of results based on strobe frequency and positions observed.

Conclusion

• Significance of understanding the difference between velocity, speed, acceleration, and their respective calculations.
• Importance of experimental setup and consideration of uncertainties in measurements.