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# 5: Two-Dimensional Kinematics

• 5.1: Prelude to Motion in Two and Three Dimensions
• 5.2: Displacement and Velocity Vectors
The position function is graphed as a vector from the origin of a chosen coordinate system to describe the position of a particle as a function of time of a particle moving in two or three dimensions. The displacement vector gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions. Instantaneous velocity is graphed as a vector that gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions.
• 5.3: Acceleration Vector
In two and three dimensions, the acceleration vector can have an arbitrary direction and does not necessarily point along a given component of the velocity. The instantaneous acceleration is produced by a change in velocity taken over a very short time period. Instantaneous acceleration is a vector in two or three dimension which can be found by taking the derivative of the velocity function with respect to time.
• 5.4: Projectile Motion
Projectile motion is the motion of an object subject only to the acceleration of gravity, where the acceleration is constant, as near the surface of Earth. To solve projectile motion problems, we analyze the motion of the projectile in the horizontal and vertical directions using the one-dimensional kinematic equations for x and y.
• 5.5: Uniform Circular Motion
Uniform circular motion is motion in a circle at constant speed. Centripetal acceleration is the acceleration pointing towards the center of rotation that a particle must have to follow a circular path. Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. An object executing uniform circular motion can be described with equations of motion.
• 5.6: Relative Motion in One and Two Dimensions
When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified. Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame. If two reference frames are moving relative to each other at a constant velocity, then the accelerations of an object as observed in both reference frames are equal.
• 5.7: Motion in Two and Three Dimensions (Exercises)
• 5.8: Motion in Two and Three Dimensions (Summary)
• 5.9: Motion in Two Dimensions
An object moving with constant velocity must have a constant speed in a constant direction.
• 5.10: Vectors Revisited
Vectors are geometric representations of magnitude and direction and can be expressed as arrows in two or three dimensions.
• 5.11: Projectile Motion Revisited
Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory.
• 5.12: Multiple Velocities
Relative velocities can be found by adding the velocity of the observed object to the velocity of the frame of reference it was measured in.