Physics 8A and B

Scalar quantity

A physical measurement that is fully described by its magnitude (size or amount) alone, without any associated direction. Examples include temperature, mass, and time.

Vector quantity

A physical quantity that possesses both magnitude (size or amount) and direction. Vectors are often represented visually by arrows, where the length of the arrow signifies the magnitude and the arrowhead indicates the specific direction.

How can the direction of a vector be communicated?

The direction of a vector can be clearly communicated using several methods: cardinal directions (North, East, South, West), relative terms like left/right, or specific angles (e.g., 30^{\circ} above the horizontal). For vectors constrained to one dimension, positive ( + ) and negative ( - ) signs are typically used to denote opposing directions.

Distance (d)

A scalar quantity that represents the total length of the path traveled by an object, regardless of changes in direction. It is always a non-negative value and is conventionally measured in meters (m).

Displacement (s)

A vector quantity that represents the shortest straight-line distance from an object's initial position to its final position, including the direction of that straight line. It is measured in meters (m).

How are one-dimensional (1D) vectors combined?

For one-dimensional motion, vectors are combined by simply adding their magnitudes algebraically, taking into account their respective positive or negative directions. Conceptually, this is like combining arrows head-to-tail along a single line to determine the total displacement.

How can vectors be represented graphically?

Vectors can be represented graphically in a two-dimensional Cartesian coordinate system using x and y axes, where their components are specified. Alternatively, they can be described by their magnitude and an angle relative to a reference axis. When multiple vectors are involved, they can be combined using vector addition rules to find a resultant vector.

Velocity

A vector quantity defined as the rate at which an object changes its displacement, or displacement per unit time. It indicates both the speed and the direction of motion. Velocity is typically measured in meters per second (m/s). To convert velocity from kilometers per hour (km/h) to meters per second (m/s), you divide by 3.6 (since 1 \text{ km} = 1000 \text{ m} and 1 \text{ h} = 3600 \text{ s}). Conversely, to convert from meters per second (m/s) to kilometers per hour (km/h), you multiply by 3.6.

Acceleration

A vector quantity defined as the rate of change of an object's velocity over time. It measures how quickly an object's velocity (magnitude or direction, or both) changes. Acceleration is measured in meters per second squared (m/s^2). It occurs whenever there is a change in speed (speeding up or slowing down) or a change in direction (even if speed is constant). When an object speeds up, it is said to be accelerating in the direction of its motion. When an object slows down, it is decelerating, meaning its acceleration is in the opposite direction of its motion.

Displacement-Time Graph

A graph that plots an object's displacement against time. The gradient (slope) of a displacement-time graph represents the object's average velocity over that time interval.

Velocity-Time Graph

A graph that plots an object's velocity against time. The gradient (slope) of a velocity-time graph represents the object's acceleration. For an object traveling at a constant speed, its acceleration is 0, resulting in a zero gradient. The area under the velocity-time graph between two time intervals represents the total displacement of the object. A non-zero constant acceleration will result in a linear velocity-time graph (a straight line with a constant slope).

Speed-Time Graph

A graph that plots an object's speed against time. Any point on the graph represents the instantaneous speed at that particular moment. The area under the speed-time graph between two time intervals represents the total distance traveled by the object. The gradient (slope) between two points on the graph represents the average acceleration over that interval.

Acceleration-Time Graph

A graph that plots an object's acceleration against time. For constant acceleration, the gradient of this graph is 0. In this case, the average acceleration is equal to the instantaneous acceleration and is also given by the gradient of the corresponding velocity-time graph. The area under the acceleration-time graph between two time intervals represents the change in velocity of the object during that period.

Vertical Motion and Gravity

Motion in the vertical direction is influenced by gravitational acceleration, approximately 9.8 \, m/s^2 (denoted as g) downwards. When an object is moving upwards, its velocity is in the opposite direction to acceleration due to gravity. When an object is moving downwards, its velocity and acceleration due to gravity are in the same direction.