How does the choice between integrating signed velocity and absolute velocity differentiate net displacement from total path length?

Conditions

• Motion occurs along a 1D line
• Velocity $v(t)$ changes sign

Reasoning, step by step

1. Define velocity $v(t)$ as the rate of change of position, carrying a sign indicating direction.
2. Compute the definite integral $\int_{t_0}^{t_1} v(t) dt$. Positive areas cancel negative areas.
3. Interpret the result as the final position minus initial position (Displacement).
4. Compute the definite integral $\int_{t_0}^{t_1} |v(t)| dt$. All areas are treated as positive.
5. Interpret the result as the sum of lengths of all segments traversed (Total Distance).

Example

The script notes: 'Integrating signed velocity gives net displacement, while integrating its absolute value gives total distance traveled.'

Common misconceptions

• Assuming displacement and distance are always equal.
• Thinking that a zero integral of velocity implies no movement occurred (it could mean returning to start).

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Explore next

• In what scenario would displacement be zero but distance be large?
• How does this concept extend to vector-valued functions in higher dimensions?