The Cartesian plane is a graphical representation for equations used in math and science. It consists of two perpendicular number lines, called the x-axis and y-axis, intersecting at a zero point called the origin. The x-axis extends horizontally to the right and left of the origin, while the y-axis extends vertically above and below it. Values above, and to the right of the origin are positive, and values below, and to the left of the origin are negative. Measurements in physics are directional, so the cartesian plane is used to orient the direction of travel of an object or particle. Conventionally, if something is moving up or to the right, the motion is positive, and if it is moving down or to the left, the motion is negative. It’s important to understand that positive and negative signs do not indicate bigger or smaller numbers in physics, they indicate the direction the object is moving in reference to a Cartesian plane.
Vectors vs. Scalars
We’ve talked a lot about the direction of an object as indicated by it’s sign. However, how fast or slow something is moving (magnitude) is also part of its motion and has a numerical value. Measurements that describe both magnitude AND direction are called vectors. Vectors are often represented graphically as arrows, where the length of the arrow corresponds to the magnitude of the vector, and the direction of the arrow indicates the direction in which the vector acts. Examples of vectors in physics are velocity, acceleration, and force.
A scalar is a quantity that is solely characterized by its magnitude, or size, and does not have any direction associated with it. Examples of scalar quantities include mass, temperature, speed, and energy. Scalar measurements do not include a sign.
Distance vs. Displacement
Distance and displacement are both terms that you’ve probably used before, but they have distinct meanings and implications in physics. Distance refers to the total length of the path traveled by an object, irrespective of its direction. It is a scalar quantity, and is always positive or zero. For example, if a person walks from point A to point B and then back to point A along a curved path, the total distance traveled is the sum of the lengths of the two segments of the path.
Displacement, on the other hand, is a vector quantity that refers to the change in position of an object and is characterized by both magnitude and direction. It is simply the difference between the final and initial positions of an object regardless of the path taken to get there. In the example of walking from point A to point B and back to point A, the displacement is zero because the person ends up at the same position where they started.
Speed vs Velocity
Speed and velocity are both measures of how fast an object is moving, but they differ in their conceptual meanings and mathematical representations.
Speed is a scalar quantity that refers to the rate at which an object covers distance. It is calculated by dividing the distance traveled by the time taken. Speed only indicates how fast an object is moving, and does not take into account the direction of motion.
Velocity, on the other hand, is a vector quantity that specifies not only how fast an object is moving, but also in which direction it is moving. It is the rate of change of displacement with respect to time, and can be calculated by dividing the displacement by the time taken. Unlike speed, velocity can be positive, negative, or zero, depending on the direction of motion. For example, if a car travels 100 kilometers eastward in 2 hours, its velocity is +50 km/h east. If the car travels 100 kilometers westward in 2 hours, its velocity is -50 km/h (negative to indicate the opposite direction).