motion in a plane

CHAPTER THREE: MOTION IN A PLANE

3.1 INTRODUCTION

• 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.

3.2 SCALARS AND VECTORS

• 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).

3.2.1 POSITION AND DISPLACEMENT VECTORS

• 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.

3.2.2 EQUALITY OF VECTORS

• 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.

3.3 MULTIPLICATION OF VECTORS BY REAL NUMBERS

• 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.

3.4 ADDITION AND SUBTRACTION OF VECTORS — GRAPHICAL METHOD

• 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.

3.5 RESOLUTION OF VECTORS

• 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.

3.6 VECTOR ADDITION — ANALYTICAL METHOD

• 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.

3.7 MOTION IN A PLANE

• 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.

3.8 MOTION IN A PLANE WITH CONSTANT ACCELERATION

• 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

3.9 PROJECTILE MOTION

• 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.

3.10 UNIFORM CIRCULAR MOTION

• 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.

SUMMARY

• Scalar vs. Vector definitions and examples,

• Vector operations, addition, and properties.

• Motion equations and principles for projectile and circular motion.

POINTS TO PONDER

1. Difference between path length and displacement.

2. Relationship between average speed and average velocity.

3. Independence of x and y motion in two-dimensional motion analysis.