Understanding Motion: A Guide to Cartesian Coordinate Systems & Kinematics

Posted on Nov 9, 2024 in Physics

Cartesian Coordinate System

Frame of Reference

A frame of reference is defined by a point in space, called an origin, and a way of locating an object relative to the origin. In a Cartesian coordinate system, the way to locate an object is based on a set of perpendicular axes that intersect at the origin. To locate an object in space, we need a frame of reference with 3 axes. This year, however, we will study movement in a plane, and 2 axes will be enough.

Position

In a Cartesian coordinate system, an object’s position is determined by its coordinates, which represent the distance from the origin along each axis.

Path

Given a frame of reference, we say that an object is in motion when its position varies over time. The path, or trajectory, is the line made from a series of successive positions of a body in motion. Depending on the shape of the path, a given motion is classified as:

• Linear: when it is a straight line
• Curvilinear: if not a straight line
• Circular: in the special case of movement along a circumference

It’s worth noting that an object’s path, just like its position, depends on the frame of reference.

Position Vector (r->)

The position vector has its initial point at the origin of the coordinate system, and its endpoint is the position of the object. The magnitude of the vector is the same as the linear distance between the object and the origin of the frame of reference. When the object is at rest in a frame of reference, the position vector does not change, and when the object does move, its position vector changes, and its endpoint follows the object’s path.

Displacement Vector (Δr->)

The displacement vector, Δr, has its initial point at the starting point and its endpoint at the final point.

Distance Traveled

When an object moves without changing direction, the distance traveled is the distance measured along the path between the object’s initial location and its final location.

To determine the position along a path, we specify an origin and measure the distance relative to it. If at a point in time, t₀, the object is in position s₀, and at some later time t, it is at position s, then its displacement along the path in the time interval Δt = t – t₀ is given by: Δs = s – s₀. The distance traveled is the absolute value of this displacement (d = |Δs|) and it is not necessarily the same as the magnitude of the displacement vector since the first is measured along the path and the second along a straight line.

Velocity (v->)

Velocity is the physical quantity that describes how a moving object’s position changes.

Average Velocity (v->_m)

Average velocity on a path comes from dividing the displacement vector by the time interval. v->_m = Δr-> / Δt, where Δt = t_f – t₀

Instantaneous Velocity (v->)

Instantaneous velocity specifies the velocity of an object at each instant of time.

Average Speed (c_m)

Average speed is the ratio between the distance traveled and the time required. c_m = d / Δt

Acceleration (a->)

Acceleration is the physical quantity that describes changes in the velocity of an object.

Average Acceleration (a->_m)

Average acceleration on a path is the change in velocity divided by the time interval of the change. a->_m = Δv-> / Δt, where Δv = v-> – v->₀ and Δt = t – t₀

Instantaneous Acceleration

Instantaneous acceleration is the acceleration in an instant of time.

Intrinsic Components of Acceleration

Tangential Acceleration (a->_t)

This is the component along the direction tangent to the path, the same as the velocity vector, and it tells you about changes in speed. If the tangential acceleration has the same direction as the velocity, the speed increases, and if the quantities have opposite directions, the speed diminishes.

Normal Acceleration (a->_n)

This type is also called centripetal acceleration and points perpendicular to the tangent of the path. The centripetal acceleration tells you about changes in the direction of the velocity.

Types of Motion

a_n = 0:

1. a_t = 0: Uniform Linear Motion
2. a_t ≠ 0:
   1. a_t = constant: Uniformly Accelerated Linear Motion
   2. a_t ≠ constant: Accelerated Linear Motion

a_n ≠ 0:

1. r = constant:
   1. a_t = 0: Uniform Circular Motion
   2. a_t ≠ 0:
      1. a_t = constant: Uniformly Accelerated Circular Motion
      2. a_t ≠ constant: Accelerated Circular Motion
2. r ≠ constant:
   1. a_t = 0: Uniform Curvilinear Motion
   2. a_t ≠ 0:
      1. a_t = constant: Uniformly Accelerated Curvilinear Motion
      2. a_t ≠ constant: Accelerated Curvilinear Motion

Linear Motion

Uniform Linear Motion

Both acceleration components are zero. Since acceleration is zero, the velocity remains constant, and its value will always equal the average velocity. The distance traveled is directly proportional to the time of travel. v = (x – x₀) / t, x – x₀ = v * t, x = x₀ + v * t