Kinematics: Vectors and Motion in Two Dimensions

Topic 1.5: Kinematics - Vectors and Motion in Two Dimensions

Introduction

• Welcome to Topic 1.5 focused on Kinematics: Vectors and Motion in Two Dimensions.

• This session is designated as Daily Video One, presented by Wesley Baker from Creekview High School, Carrollton, Texas.

Overview of Vectors

• Vectors: Defined as quantities that have both magnitude and direction.

  • Magnitude: Refers to the size of the vector; always associated with how much of the quantity there is.

Vectors in One Dimension

• Example: A confused dog walks forward 10 meters and then backward 7 meters.

  • Objective: Determine the dog’s displacement.

  • First vector: 10 meters forward (positive direction).

  • Second vector: 7 meters backward (negative direction).

• To find the dog’s displacement, add vectors tip to tail:

  • First vector drawn forward (10 m).

  • Second vector drawn backward from the tip of the first vector.

    • This method visually demonstrates the resultant vector.

  • Resultant vector: Extends from the tail of the first vector to the tip of the second.

Mathematical Addition of Vectors

• For numerical addition:

  • Positive 10 meters (forward) + negative 7 meters (backward) = 10 + (-7) = 3 meters.

    • Result: 3 meters in the forward direction.

Motion in Two Dimensions

• Involves movement along both the X and Y directions simultaneously.

  • Orthogonal Vectors: Vectors that are at right angles to each other.

  • The terms orthogonal and perpendicular are interchangeable.

• Vectors still follow the tip to tail addition rule, regardless of their orientation.

Example of Adding Orthogonal Vectors

• Consider two vectors: 3 meters North and 4 meters East:

  • North and East are at 90 degrees (orthogonal).

  • Begin with the 3 meters North vector, then add the 4 meters East vector tip to tail.

  • Resultant vector extends from the tail of the first to the tip of the second vector.

Finding the Resultant Vector

• To calculate magnitude of the resultant vector:

  • Use the Pythagorean theorem: $a^2 + b^2 = c^2$

    • Where $a = 3$ meters and $b = 4$ meters.

  • Result: $3^2 + 4^2 = 9 + 16 = 25$

    • Therefore, $c = ext{hypotenuse (R)} = <br>oot{25} = 5$ meters.

Direction of the Resultant Vector

• To find direction, use the arctangent function:

  • $an^{-1} ext{ (opposite/adjacent)}$

    • For our case: $an^{-1} rac{4}{3}$.

  • Ensure calculator is in degree mode.

    • Resultant angle: $53^ ext{o}$.

Practical Application: Treasure Map Example

• Given instructions starting at a palm tree:

  • 5 meters North, 10 meters East, 3 meters North, 6 meters West, 7 meters South.

• Following the instructions step-by-step adds complexity; instead, sum vectors in each direction:

  • X Direction:

    • 10 meters East (positive) + 6 meters West (negative) = $10 - 6 = 4$ meters East.

  • Y Direction:

    • 5 meters North (positive) + 3 meters North (positive) - 7 meters South (negative) = $5 + 3 - 7 = 1$ meter North.

Resultant Vector from Total Movements

• Combine total movements in X and Y:

  • Use Pythagorean theorem again:

    • $4^2 + 1^2 = 16 + 1 = 17$

    • Resultant: $<br>oot{17} ext{ meters} ext{ (around 4.12 meters)}$.

Direction to the Treasure

• Determine heading for the treasure:

  • $an^{-1} rac{1}{4}$ to find angle.

    • Resultant angle: $14^ ext{o}$.

Conclusion: Key Takeaways

• Vectors are added tip to tail for resultant calculation.

• When adding vectors at 90 degrees, utilize the Pythagorean theorem to find magnitude.

  • Example: The Pythagorean result of vectors 3 and 4 yields a resultant vector of 5.

• Use the arctangent function to deduce direction of resultant vectors.

• Understanding vectors enhances practical applications like simplifying navigation in the treasure map example.

Encouragement

• Thank you for your attention. Remember, you are doing great in physics and prepare well for the AP exam!