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Calculus: The Velocity Problem

Jul 11, 2024

Calculus: The Velocity Problem

Introduction to the Velocity Problem

  • Scenario: Driving on the highway
    • At 2:00: At the 100-mile marker
    • At 2:15: At the 110-mile marker
  • Question: How fast am I going?
    • Clarification needed: How fast at 2:00? At 2:15? Over the interval?

Average Velocity

  • Definition: Change in distance (ΔD) divided by change in time (ΔT)
    • Mathematical notation: ΔD / ΔT
  • Example Calculation:
    • ΔD = 110 miles - 100 miles = 10 miles
    • ΔT = 2:15 - 2:00 = 15 minutes
    • Average velocity in miles per minute: 10 miles / 15 minutes = 2/3 miles per minute
    • Unit conversion: 2/3 miles per minute * 60 minutes/hour = 40 miles per hour
  • Conclusion: Average velocity over 15 minutes is 40 miles per hour*

Instantaneous Velocity

  • Question: What would a cop measure at exactly 2:15?
    • Average velocity over 15 minutes: 40 miles per hour
    • Issue: No information on exact speed at 2:15

Different Scenarios:

  • Speeding between 2:00 to 2:10, then slowed down
  • Average velocity is 40 miles per hour but actual speed varies

Concept of Instantaneous Velocity

  • Instantaneous velocity: Velocity at one exact time (e.g., 2:15)
  • Comparison:
    • Average velocity: Over an interval
    • Instantaneous velocity: At a specific moment

Method to Determine Instantaneous Velocity

  • Use smaller and smaller time intervals
    • Example: Time intervals from 2:00-2:15 down to seconds
  • Calculate average velocities for each interval
    • Smaller intervals give a finer measurement
  • Hypothetical data table showing average velocities over decreasing intervals
    • As intervals approach 2:15, velocities converge to a specific value (e.g., 60 miles per hour)
  • Conclusion: Instantaneous velocity approximated by limiting process of average velocities over decreasing intervals

Practical Application: Lidar Guns

  • How it works:
    • Sends out pulses of light
    • Measures distance between cop and car at tiny time intervals
    • Calculates average velocity over these small intervals
  • Modern lidar guns use this principle to give accurate speed readings
    • Tiny intervals provide a close approximation of instantaneous velocity

Summary

  • Average velocity: Useful for overall speed over an interval
  • Instantaneous velocity: Important for specific moments
  • Limiting process: Key to approximating instantaneous velocity
  • Practical relevance: Used in speed detection technologies like lidar guns

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