Particle Motion in Calculus

Overview

• Particle motion problems are common in AP Calculus exams, testing various calculus concepts.
• Problems can be presented with position, velocity, or acceleration equations, graphs, or tables.
• May involve finding direction changes, maximum position, or speed.

Key Concepts

Position, Velocity, and Acceleration

• Ability to differentiate or integrate to move between position, velocity, and acceleration.
• Initial Value Problems: Given velocity and initial position or acceleration and initial velocity, solve differential equations.

Speed and Direction

• Speed is the absolute value of velocity.
• Velocity and acceleration with the same sign indicate increasing speed.
• Velocity and acceleration with different signs indicate decreasing speed.

Graphical Interpretation

• Speed graph: Non-directed distance from velocity graph to t-axis.
• Reflect velocity graph below t-axis for speed graph.

Distance and Displacement

• Total distance: ( \int_a^b |v(t)| , dt )
• Net distance (displacement): ( \int_a^b v(t) , dt )

Calculations

Non-Calculator Section

• Example problems with particle movement along the x-axis.

Calculator Usage

• Use a graphing calculator for velocity and acceleration computations.

Scenarios

• Initially: ( t = 0 )
• At rest: ( v(t) = 0 )
• Moving right/forward/up: ( v(t) > 0 )
• Moving left/backward/down: ( v(t) < 0 )
• Average velocity: ( \frac{x(b) - x(a)}{b - a} )
• Instantaneous velocity: ( v(c) = x'(c) )
• Acceleration: ( a(c) = v'(c) = x''(c) )
• Velocity increasing: ( a(t) = v'(t) > 0 )
• Velocity decreasing: ( a(t) = v'(t) < 0 )
• Total distance traveled: ( \int_a^b |v(t)| , dt )

Multiple Choice Sample Problems

1. Direction Change: At which ( t ) does the bug change direction?
2. Total Distance: Calculate bug's distance from ( t = 0 ) to ( t = 8 ).
3. Velocity Zero: At what ( t ) is the particle's velocity zero?
4. Maximum Acceleration: Particle's max acceleration in an interval.
5. Farthest Right Position: When is the particle farthest right?
6. Particle at Rest: Determine ( t ) when particle is at rest.

Answer Key

• Answers to multiple choice: 1C, 2B, 3C, 4D, 5B, 6E, 7B, 8C, 9D, 10C, 11B, 12C.

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