1.1 Position, Velocity, and Acceleration: Kinematics Study Guide for AP Physics 1

Understanding the core principles of position, velocity, and acceleration is critical to describing motion accurately in physics. These concepts form the building blocks for analyzing how objects move in different contexts. In this guide, we’ll explore these topics in-depth, focusing on their significance, practical applications, and how they relate to frames of reference.

Frames of Reference in Motion

Key Concept: A frame of reference provides a point of view for observing and measuring motion. For example, two people sitting in a moving car may perceive themselves as stationary relative to each other but are in motion relative to someone standing outside.

• Fixed Frames: Do not move (e.g., a classroom wall).
• Moving Frames: Constantly move (e.g., a car traveling at a constant speed).

Example:
Two students sit at their desks. They are stationary relative to each other but are moving relative to the solar system as Earth orbits the sun.

Position 🌎

Position is defined as an object’s location relative to a reference point. It can be represented using a position vs. time graph:

• Sloped Line: Indicates motion.
  • Positive Slope (/): Moving away from the detector.
  • Negative Slope (\): Moving towards the detector.
• Slope of the Graph: Represents velocity.
  • Straight Line Slope: Constant velocity.
  • Curved Line: Acceleration (changing velocity).
  • Zero Slope: Object is at rest.

Graph Tip: The y-intercept of a position-time graph represents the initial position (displacement).

Scalar vs. Vector Quantities 💫

Scalar Quantities:

• Definition: Described by magnitude alone (e.g., 5 meters).
• Examples: Distance, speed, mass.
• Representation: Scalars are straightforward numbers without a direction.

Vector Quantities:

• Definition: Have both magnitude and direction (e.g., 10 m/s east).
• Examples: Displacement, velocity, acceleration, force.
• Representation: Vectors are depicted with arrows; the length represents magnitude, and the direction shows the vector’s direction.

Key Points:

• Scalar quantities use simple arithmetic, while vector quantities require vector addition/subtraction.
• Vector quantities are denoted with arrows or bold symbols.

Displacement vs. Distance 🚴‍♀️

Displacement:

• Definition: The change in position from an initial point to a final point.
• Vector Quantity: Has direction (positive or negative).
• Unit: Meters (m).
• Example: How far you are from your starting point.

Distance:

• Definition: The total path traveled by an object.
• Scalar Quantity: No direction involved.
• Unit: Meters (m).
• Example: The length of the path taken during a journey.

Example:
A car travels 100 km along a straight road but takes a 50 km detour before reaching its destination.

• Displacement: 100 km (start to finish).
• Distance: 150 km (total path traveled).

Speed vs. Velocity 🏇

Speed:

• Definition: Rate of motion (scalar).
• Formula: $S = \frac{D}{t}$
• Unit: Meters per second (m/s).

Velocity:

• Definition: Speed with direction (vector).
• Formula: $V = \frac{x}{t}$
• Unit: Meters per second (m/s).
• Graph Representation: Velocity vs. Time Graphs
  • Above x-axis: Moving away from detector.
  • Below x-axis: Moving towards detector.
  • Slope: Represents acceleration.
  • Zero Slope: Constant velocity.

Key Tip: The area under a velocity-time graph represents displacement.

Acceleration 🚀

Acceleration measures the change in velocity over time (vector). It indicates speeding up, slowing down, or changing direction.

• Formula: $A_{\text{avg}} = \frac{\Delta V}{t}$
• Graph Representation: Acceleration vs. Time Graph
  • Y-Intercept: Initial acceleration.
  • Zero Slope: Constant acceleration.
  • Area under the curve: Represents change in velocity.

Example:
Calculate the velocity of an object from 0s to 7s using an acceleration-time graph (assuming constant rates and provided values). Break the graph into geometric shapes to find the area and sum them to find the total velocity change.

Motion Summary Table

Practice Problems

1. Problem 1: Calculate the final velocity of a car accelerating from rest at $3 \, \text{m/s}^2$ for 5 seconds.
   Solution:

   $$V_f = V_o + at \Rightarrow V_f = 0 + (3)(5) = 15 \, \text{m/s}$$
   

2. Problem 2: An object is thrown upward at $20 \, \text{m/s}$. Calculate its maximum height.
   Solution:
   Use $V_f^2 = V_o^2 + 2ay$.
   Given $V_f = 0 \, \text{m/s}$, $a = -9.8 \, \text{m/s}^2$ and $V_o = 20 \, \text{m/s}$:

   $$0 = 20^2 + 2(-9.8)y \Rightarrow y = 20.4 \, \text{m}$$