AP Calculus: Particle Motion – Accumulation, Net Distance, Total Distance

Particle Motion – Accumulation, Net Distance, Total Distance

Concept Summary

• Position function s(t): The location of a particle at time t, given as a differentiable function; its derivative is velocity v(t) = s'(t).
• Velocity v(t): The instantaneous rate of change of position, s'(t); sign indicates direction (positive = forward, negative = backward).
• Accumulation (displacement): The net change in position from t = a to t = b, calculated as ? v(t) dt = s(b) – s(a).
• Net distance: Synonymous with displacement; the straight-line distance from start to end, regardless of path taken.
• Total distance: The sum of all distances traveled, calculated as ? |v(t)| dt; always non-negative.

Core Questions

WHAT (definitional)

Q: What is displacement? A: The net change in position, ? v(t) dt, which can be positive, negative, or zero. Trap/Clarification: Displacement-distance traveled; it ignores direction reversals.

Q: What is total distance? A: The sum of all distances traveled, ? |v(t)| dt, always-0. Trap/Clarification: Total distance requires absolute value; omitting it gives displacement, not distance.

WHY (causal/explanatory)

Q: Why is the sign of v(t) important? A: The sign determines direction: positive v(t) increases position, negative v(t) decreases it. Trap/Clarification: A particle can be moving backward (v(t) < 0) while its position s(t) is still positive.

Q: Why is total distance always-displacement? A: Total distance accounts for all motion (including reversals), while displacement only measures net change. Trap/Clarification: If a particle reverses direction, displacement undercounts the actual path length.

HOW (process/application)

Q: How do you calculate displacement from v(t)? A: Integrate v(t) over the interval: ? v(t) dt = s(b) – s(a). Trap/Clarification: Do not take the absolute value; displacement can be negative.

Q: How do you find total distance from v(t)? A: Integrate the absolute value of v(t): ? |v(t)| dt, or split the integral at direction changes. Trap/Clarification: Forgetting to split the integral at v(t) = 0 leads to incorrect results.

Q: How do you determine when a particle changes direction? A: Find where v(t) = 0 and check sign changes (e.g., v(t) crosses the t-axis). Trap/Clarification: A v(t) = 0 point is not always a direction change (e.g., v(t) = t² touches but doesn’t cross).

CAN (conditions/possibilities)

Q: Can displacement be zero while total distance is positive? A: Yes, if the particle returns to its starting position (e.g., oscillates). Trap/Clarification: Zero displacement does not imply the particle was stationary.

Q: Can total distance ever equal displacement? A: Yes, if the particle never changes direction (i.e., v(t) does not change sign). Trap/Clarification: This requires v(t)-0 or v(t)-0 for the entire interval.

Quick Facts & Traps

• Fact: s(t) is an antiderivative of v(t); v(t) is the derivative of s(t).
• Trap: Confusing s(t) with distance-Reality: s(t) is position, which can decrease.
• Fact: Total distance =-|v(t)| dt = sum of distances between direction changes.
• Trap: Integrating v(t) without absolute value-Reality: Gives displacement, not total distance.
• Fact: Direction changes occur where v(t) = 0 and the sign of v(t) flips.
• Trap: Assuming v(t) = 0 always means a direction change-Reality: Check for sign change (e.g., v(t) = t³ at t = 0).

Rapid-Fire True/False

• Statement: If v(t) > 0 for all t in [a, b], then displacement = total distance. Answer: TRUE Why the common mistake happens: Students forget that total distance = displacement only when v(t) doesn’t change sign.

• Statement: A particle with s(2) = 5 and s(5) = 3 has a displacement of 2. Answer: FALSE (Displacement = s(5) – s(2) = –2) Why the common mistake happens: Misapplying the order of subtraction (s(final) – s(initial)).

• Statement: If v(t) = 0 at t = 1, the particle must change direction there. Answer: FALSE Why the common mistake happens: Overlooking cases where v(t) touches zero without crossing (e.g., v(t) = (t – 1)²).