Kinematics in two-dimensional surface

This is section one of the course in classical mechanics available on Physics 12: Classical Mechanics and Grade-12 Physics: Mechanics. In addition to the kinematics itself, uniform motion and uniformly accelerated motion are discussed in this course. Problem solving tutorials are also included to the course. You will find complete discussion of each topic at Grade-12 Physics: Mechanics or Physics 12: Classical Mechanics.

Kinematics

Kinematics is the study of the motion without contribution of the forces. In this lecture I will find kinematics equations of motion of an object that is moving on a flat two-dimensional surface, where you can define a rectangular two-dimensional coordinate system with 'X' and 'Y' axises and origin 'O', and with constant unit vectors î and ĵ, on X, and Y axises respectively (fig 1).

fig 1: Position vector of an object in XY-coordinate system with

unit vectors î and ĵ on X, and Y axises.

Position vector of the object at any given time, on this coordinate system, is a vector from the origin of the coordinate system to the position of the object at that time; and you can represent it by:

The components of the position vector on X and Y axises can be found by drawing two straight lines from the position of the object, perpendicular to the X and Y axises; the intersecting points of these two lines with X and Y axises, are X and Y components of the position vector. So the position vector in terms of its components can be written as:

While the object is moving on this two-dimensional plane, its position vector changes over time.

fig 2: While the object is moving on the plane, its position vector varies over time.

So the average velocity of the object in time interval of Δ t = t' − t , is defined by:

and on a rectangular two-dimensional coordinate system with X and Y axises, it can be written as:

The derivative of the position vector with respect to time, is the instantaneous velocity (or simply the velocity) of the object at any given time.

Velocity in terms of its components on a rectangular two-dimensional coordinate system, can be written as:

The average acceleration of the object in time interval of Δ t = t' − t , is also defined by:

and in terms of its components on a rectangular two-dimensional coordinate system, it can be written as:

The derivative of the velocity with respect to time is also the instantaneous acceleration ( or simply the acceleration ) of the object at any given time:

and in terms of its components on a rectangular two-dimensional coordinate system, it can be written as:

Uniform Motion

Uniform motion is a motion with constant velocity:





So at a given time ' t ', the position vector of the object can be found as:





and its components on two-dimensional surface, on XY- coordinate system, can be written as:









Uniformly Accelerated Motion





Uniformly accelerated motion is a motion with constant acceleration:

So at a given time ’t’ the velocity in terms of time can be written as:

On two-dimensional surface its components are:





The components of velocity in terms of position, on two-dimensional surface can be found as:

And the position vector at a given time ’t’ can be written as:

with components of:

on a two-dimensional surface.

Practice

questions and solutions regarding differentiation, limits,

integrals , continuity of functions, and

finding stationary values on the curve are available here

Problem Solving Lectures in Kinematics

Problem 1: An object is in uniform motion. Find its position vector in terms of time, from the definition of average velocity.

Problem Solving Lecture in Kinematics: 1

Problem 2: How to draw position-time diagram, find average velocity, and instantaneous velocity and acceleration from equation of motion.

Problem Solving Lecture in Kinematics: 2

Problem 3: How to find velocity in terms of position vector in uniformly accelerated motion.

Problem Solving Lecture in Kinematics: 3

Problem 4: A truck is stopped at a stoplight. When the light turns green it accelerates at 2 meters per squared second. At the same instant a car moving with speed of 36 km/h passes the truck . When and where does the truck catch up with the car?

Problem Solving Lecture in Kinematics: 4