motion in a plane

CHAPTER THREE: MOTION IN A PLANE

Previous chapter covered concepts of position, displacement, velocity, and acceleration for one-dimensional motion.
Motion in two or three dimensions requires the use of vectors.
Key questions:
- What is a vector?
- Methods for vector operations: addition, subtraction, multiplication by real numbers.
Motion in a plane includes:
- Constant acceleration.
- Projectile motion.
- Uniform circular motion.
Equations derived in the chapter can be used for three-dimensional motion as well.

Scalar Quantities:
- Defined by magnitude only (e.g., mass, speed, distance).
- Combine using ordinary algebra (add, subtract, multiply).
Vector Quantities:
- Defined by both magnitude and direction (e.g., displacement, velocity, acceleration).
- Represented in bold type or with an arrow over a letter.
- Combine using specific vector algebra rules (triangle law or parallelogram law).

Choose an origin point (e.g., O) as reference.
Position of object at time t represented by position vectors (OP = r).
Displacement vector (PP') illustrates movement from one position to another.
Displacement is direction-independent and considers only initial and final points, not path.

Two vectors A and B are equal if they have the same magnitude and direction.
Vectors can be shifted parallel to themselves without changing their properties—these are free vectors.
Localized vectors differ by their specific point of application.

Positive Multiplication: A * λ (λ > 0) keeps the direction of vector A.
Negative Multiplication: A * λ (λ < 0) reverses the direction of vector A.
Dimension of the resultant vector when multiplied by a scalar carries the dimensions of both.

Vectors are added graphically using the head-to-tail method or parallelogram method.
Commutative Property: A + B = B + A.
Associative Property: (A + B) + C = A + (B + C).
The resultant of A and -A (equal and opposite vectors) is a null vector.

A vector can be expressed as a combination of two components along specified directions:
- A = λa + µb,
- λ and µ are scalars.
Unit Vectors are defined for each axis (i.e., i, j, k) and have a magnitude of 1.

Vectors can be added component-wise:
- For vectors A and B located in the x-y plane: R = A + B → Rx = Ax + Bx, Ry = Ay + By.
Allows for easier computations than graphical methods.

Describing motion using position and displacement vectors in the x-y coordinate system.
Average velocity determined from displacement over time.
Instantaneous velocity is the limiting value of average velocity as time approaches zero.

Covers the motion of objects with constant acceleration in two dimensions.
Updates to velocity and position models are based on vector principles:
- v = v0 + at
- r = r0 + v0t + (1/2)at^2

Analyzes motion of objects projected into the air, considering horizontal (constant velocity) and vertical (accelerated) motions separately.
Parabolic path is characteristic due to gravitational effects.

Defined by constant speed along a circular path.
Acceleration is directed towards the center of the circle (centripetal acceleration).
Relationships exist between linear speed, angular speed, radius, and frequency.