Vectors in Physics

Vector Addition and Properties

Vectors are quantities that have both magnitude and direction. In physics, vectors are used to represent quantities such as displacement, velocity, and force.

• Vector Addition: When two vectors A and B are added, the resultant vector C is found using the triangle or parallelogram method.

• Pythagorean Theorem for Perpendicular Vectors: If A and B are at right angles, the magnitude of C is given by:

• Zero Vector: The sum of two vectors is the zero vector only if they are equal in magnitude and opposite in direction (antiparallel).

• Parallel and Antiparallel Vectors: Vectors in the same direction are parallel; in opposite directions, they are antiparallel.

Example: If two vectors are added and the result is the largest possible magnitude, they must be in the same direction.

Vector Components

Any vector can be broken into components along the x and y axes. The number of nonzero components depends on the vector's orientation.

• Component: The projection of a vector along an axis.

• Example: A vector in the x-y plane may have one or two nonzero components depending on its direction.

Motion in One Dimension

Displacement, Distance, and Speed

Motion along a straight line can be described using displacement, distance, velocity, and speed.

• Displacement: The straight-line distance from the initial to the final position, with direction.

• Distance: The total length of the path traveled, regardless of direction.

• Average Velocity: Displacement divided by total time.

• Average Speed: Total distance divided by total time.

Example: A runner travels 1600 m east, then halfway back (800 m), in 5 minutes (300 s): Displacement = 800 m (east), so m/s (east). Total distance = 2400 m, so avg speed = m/s.

Worked Example: Cat on a Line

• Average Velocity:

• Average Speed:

• Example Calculation: If m, m, s: m/s

Acceleration

Definition and Types

Acceleration is the rate of change of velocity with respect to time. It can be positive (increasing speed) or negative (decreasing speed, also called deceleration).

• Average Acceleration:

• Instantaneous Acceleration: The acceleration at a specific instant.

• Scalar vs. Vector: Acceleration is a vector quantity; it has both magnitude and direction.

Examples of Acceleration

• Increasing Velocity: A car speeding up down a hill.

• Decreasing Velocity (Deceleration): A car slowing down as it approaches a stop sign.

• Worked Example: A drag racer slows from 28 m/s to 13 m/s in 3 seconds: m/s2

Constant Acceleration and Kinematic Equations

Constant Acceleration

When acceleration is constant, motion can be described by a set of kinematic equations.

• Variables:

  Variable	Units
  x	Meters (m)
  v	Meters per second (m/s)
  t	Seconds (s)
  a	Meters per second squared (m/s2)

• Kinematic Equations:

  where is initial velocity, is final velocity, is acceleration, is time, and is displacement.

Example: A car accelerates from rest at 10 m/s2 for 3 seconds: m/s

Freely Falling Bodies

Gravity and Free Fall

Objects in free fall experience constant acceleration due to gravity, denoted as .

• Acceleration due to Gravity: m/s2 (often approximated as 10 m/s2 in calculations).

• Direction: Downward, toward the center of the Earth.

• Independence from Mass: All objects fall with the same acceleration in the absence of air resistance.

• Kinematic Equations for Free Fall: Use the same equations as above, with (if upward is positive).

Example: An object dropped from rest falls for 3 seconds: m (downward)

Summary Table: Kinematic Variables and Units

Additional info: These notes cover foundational concepts in introductory kinematics and vector analysis, suitable for first-year college physics students. The equations and examples provided are essential for solving problems involving motion in one dimension, vector addition, and free fall.