AP Physics 1: Scalars, Vectors, and 1D Motion - Your Ultimate Review 🚀

Hey there, future AP Physics champ! Let's break down scalars, vectors, and one-dimensional motion. This is your go-to guide for acing the exam, especially the night before when every minute counts. Let’s get started!

Scalars and Vectors: The Basics

• Scalars: These are quantities that are fully described by their magnitude (size) alone. Think of them as just numbers with units. 📏
  • Examples: Distance, speed, time, mass, temperature.
• Vectors: These quantities are described by both magnitude and direction. 🧭
  • Examples: Position, displacement, velocity, acceleration, force.

Vector Representation

• Vectors are often represented by arrows. The length of the arrow shows the magnitude, and the arrow points in the direction of the vector.
• Vector quantities include:
  • Position: Where an object is located.
  • Displacement: Change in position (how far and in what direction).
  • Velocity: Rate of change of position (speed + direction).
  • Acceleration: Rate of change of velocity.
• Notation: A vector is often denoted with an arrow above the symbol, like $\vec{v}$ for velocity. In one dimension, we often drop the arrow and use a + or - sign to indicate direction.

Examples of Scalars and Vectors

• Scalars:
  • Distance traveled during a trip: 300 miles
  • Speed of a car on the highway: 65 mph
• Vectors:
  • Displacement from starting point to ending point: 50 km east
  • Velocity of a plane: 500 mph, 30° north of east
  • Acceleration due to gravity: -9.8 m/s² (downward)

Vector Sum in One Dimension

Opposite Directions and Signs

• In one dimension, we use positive (+) and negative (-) signs to show direction. ➕➖
  • Rightward or upward is typically positive.
  • Leftward or downward is typically negative.
• To find the vector sum (also called the resultant vector), you add the signed magnitudes of the individual vectors.

Vector Addition Examples

• Same Direction: Two vectors pointing in the same direction: 3 m/s + 5 m/s = 8 m/s
• Opposite Directions: Two vectors pointing in opposite directions: 5 m/s - 3 m/s = 2 m/s

Practice Question

Multiple Choice Questions

1. A car travels 20 km east and then 30 km west. What is the car's total displacement? (A) 10 km east (B) 10 km west (C) 50 km east (D) 50 km west

2. A ball is thrown vertically upward with a velocity of 15 m/s. What is its velocity at the highest point? (A) 0 m/s (B) 9.8 m/s downward (C) 15 m/s upward (D) 15 m/s downward

3. Which of the following is a scalar quantity? (A) Force (B) Velocity (C) Acceleration (D) Speed

Free Response Question

A student is conducting an experiment to study the motion of a toy car on a straight track. The car starts from rest and accelerates uniformly for 3 seconds, reaching a velocity of 6 m/s. It then continues at this constant velocity for another 5 seconds before decelerating uniformly to a stop over 2 seconds.

(a) Sketch a velocity vs. time graph for the motion of the car. (3 points) (b) Calculate the acceleration of the car during the first 3 seconds. (2 points) (c) Calculate the displacement of the car during the first 3 seconds. (2 points) (d) Calculate the total displacement of the car for the entire 10 seconds. (3 points)

Scoring Guide:

(a) Velocity vs. Time Graph (3 points)

• 1 point for a straight line with a positive slope from (0,0) to (3,6).
• 1 point for a horizontal line from (3,6) to (8,6).
• 1 point for a straight line with a negative slope from (8,6) to (10,0).

(b) Acceleration during the first 3 seconds (2 points)

• 1 point for using the correct formula: $a = \frac{\Delta v}{\Delta t}$
• 1 point for the correct answer: $a = \frac{6 m/s - 0 m/s}{3 s} = 2 m/s^2$

(c) Displacement during the first 3 seconds (2 points)

• 1 point for using the correct formula: $\Delta x = v_0t + \frac{1}{2}at^2$ or area under the graph
• 1 point for the correct answer: $\Delta x = 0 + \frac{1}{2}(2 m/s^2)(3 s)^2 = 9 m$

(d) Total displacement (3 points)

• 1 point for calculating the displacement during the constant velocity phase: $\Delta x = (6 m/s)(5 s) = 30 m$
• 1 point for calculating the displacement during deceleration: $\Delta x = (6 m/s)(2 s) + \frac{1}{2}(-3 m/s^2)(2 s)^2 = 6 m$
• 1 point for the correct total displacement: 9 m + 30 m + 6 m = 45 m

Final Exam Focus

• High-Priority Topics: Focus on vector addition in one dimension, understanding the difference between scalar and vector quantities, and applying these concepts to kinematic problems.
• Common Question Types: Expect questions that involve calculating displacement, velocity, and acceleration using vector addition. Be prepared to interpret graphs and solve problems involving motion in one dimension.
• Time Management: Quickly identify whether a quantity is a scalar or a vector. Use the + and - signs to your advantage in 1D problems. Do not spend too much time on any single question.
• Common Pitfalls: Watch out for sign errors! Remember that negative signs indicate direction, not a reduction in magnitude. Always include units in your answers.