# Motion In One and Two Dimensions

Historically, the term mechanics was first used by Sir Isaac Newton to mean the science of machines and the art of making them. Today we use the word to mean the branch of science that describes and predicts the conditions of rest or motion under the effect of a set of forces. Mechanics can be divided into two parts: dynamics and statics. Dynamics is the study of motion of a body under one or more forces. Statics is the study of the condition of rest of a body under a number of forces. Dynamics is further divided into kinematics and kinetics. Kinematics is that part of dynamics which deals with motion ignoring the forces that cause it or the properties of the body in motion. Presently, we shall deal with kinematics here. Kinetics is that part which relates the motion of a body to its mass and the causal force(s). Newton's laws of motion , which are a fundamental part of kinetics, will be dealt next.

A body is a portion of matter which extends over a certain region of space and has a definite shape and size. But the study of motion of a body can be made simpler by ignoring its size and shape, and, assuming that the whole mass of the body is concentrated at a point occupying a definite position in space. In that event, we refer to the body as a particle. The particulate view of body is valid in many practical situations. The size of the earth and the sun are huge in absolute terms, but both are much smaller than the distance between them. Therefore, when we study the motion of the earth round the sun, we can treat both of them as particles!

When we observe a nearby object on the earth's surface, we intuitively determine whether it is at rest or in motion with respect to the ground. But considering the fact that the ground itself is in motion (earth rotates about its own axis and, simultaneously, revolves round the sun), the difference between rest and motion is not as clear-cut as one may think. The early scientists who searched for an absolutely fixed body met with little success. It was Albert Einstein who finally suggested that there is nothing called absolute rest or absolute motion. Rest and motion are purely relative concepts, and the significant thing is whether the body under investigation is at rest or in motion with respect to its local surroundings. These local surroundings are known as the frame of reference.

The frame of reference is usually pictured in terms of a three-dimensional rectangular coordinate system, often called cartesian coordinate system, consisting of three mutually perpendicular axes intersecting at a point. The axes are designated as x-, y- and z-axes; the point of intersection, marked by the letter O, is known as the origin of the system.

The motion of rigid bodies can be of two types: translational motion and rotational motion. When a rigid body performs translational motion, all particles constituting the body travel with the same velocity and acceleration at any given instant. Hence the paths followed by them have the same length and are parallel to each other. If these paths are straight lines, we say the body is in rectilinear motion (that is, motion in a straight line). Example: motion of a car along a flat, straight, narrow road. If the paths of the particles are curved, the body is said to be in curvilinear motion. An interesting example is the motion of a wooden horse on a merry-go-round.

When a rigid body performs rotational motion, the particles of the body located at different radial distances from the axis of rotation travel with different velocities and accelerations at any given instant. Hence the paths followed by them have different lengths. Example: motion of a fan blade or a spinning wheel.

The motions we see around us are often a combination of translation and rotation. The rolling motion of a sphere on the floor is one such example.

Presently, we shall study translational motion in one dimension and in two dimensions. The study of pure rotational motion and rolling motion will be done later within Mechanics. Here are some terms associated with translational motion. Suppose a particle which is at position `P1` at time `t1` moves to a new position `P2` at time `t2` following a curved path `P1P3P2` . The length of the actual path gives the distance travelled by the particle. Distance is a scalar quantity without direction. The displacement, on the other hand, is given by the straight line `P1P2` which joins the initial and final positions of the particle. Displacement is a vector quantity. The magnitude of displacement is the length of the straight line `P1P2` , and its direction is from `P1` to `P2` . Both distance and displacement have the same dimension of length, [L], and have the same SI unit metre (m).

The speed of a moving particle is the distance travelled by it in unit time. Speed is a scalar quantity. The velocity of a particle is its displacement per unit time. Velocity is a vector quantity. Both speed and velocity have the same dimensions, [LT -1] , and have the same SI unit metre per second (m/s).

The acceleration of a particle is the rate of change in velocity with time. The change in velocity may be due to a change in its magnitude, direction or both. Like displacement and velocity, acceleration is a vector quantity. The dimensions of acceleration are [LT-2], and its SI unit is metre per second square (m/s2). Negative acceleration is known as deceleration or retardation.

Graphical analysis of rectilinear motion, as the name suggests, is the visualisation of a particle's straight-line motion with the help of a graph. Two commonly plotted graphs are position versus time graph and velocity versus time graph.

The simplest example of accelerated motion is rectilinear motion with constant acceleration. It is possible to develop a useful set of kinematic equations either by graphical method or by the method of calculus (integration). These equations involve quantities such as displacement, time, initial and final velocities, and acceleration. The vertical motion of a freely falling body under gravity is a case of rectilinear motion with constant acceleration. A more complicated case is rectilinear motion with variable acceleration. When a body moves back and forth about a point in simple harmonic motion (a special case of periodic motion), its acceleration varies as a function of its displacement. Problems of this type will also be dealt with in due course.

The kinematic equations developed for rectilinear motion can be suitably modified to analyse motion in a plane, that is, motion in two dimensions. If a particle is thrown obliquely in air near the earth's surface, it follows a curved path that lies in a vertical plane. The particle is called a projectile, and the path followed by it is called a trajectory. A bullet fired from a gun or a ball hit by a cricket bat is an example of a projectile. Assuming acceleration due to gravity, g, remains constant throughout and air resistance is negligible, projectile motion is an example of motion in a plane with constant acceleration.

Other examples of two-dimensional motion are circular motion and motion of a boat in a river.

The relative velocity of a body with respect to another body is defined as the rate of change of position of the first body as observed by the second body. It is given by the vector difference between their absolute velocities. Similarly, the relative acceleration of a body with respect to another body is defined as the rate of change of velocity of the first body as observed by the second body. This is given by the vector difference between their absolute accelerations.

#### Basic level videos

Rest And Motion 56:19 Basic
300 5
Some Terms Associated With Translational Motion 1:18:15 Basic
350 7.5
Graphical Analysis Of Rectilinear Motion 57:17 Basic
300 5
Rectilinear Motion With Constant Acceleration 31:45 Basic
200 3
Analysis Of Rectilinear Motion By Calculus 30:46 Basic
200 3
Kinematic Equations By Graphical Method 23:29 Basic
200 3
Kinematic Equations By Integration 28:46 Basic
200 3
Problems On Kinematic Equations 1:02:21 Basic
350 7.5
Rectilinear Motion Under Gravity 50:08 Basic
300 5
Problems On Rectilinear Motion Under Gravity 27:35 Basic
200 7.5
Projectile Motion 1:02:33 Basic
350 7.5
Motion Of A Boat In A River 58:10 Basic
300 5
Relative Velocity And Relative Acceleration 38:57 Basic
200 3
Problems On Relative Velocity And Relative Acceleration 1:53:39 Basic
400 8

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-Level Problems On Motion In One And Two Dimensions I 1:05:48
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Advanced-Level Problems On Motion In One And Two Dimensions II 1:05:03
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Advanced-Level Problems On Motion In One And Two Dimensions III 1:11:12
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Advanced-Level Problems On Motion In One And Two Dimensions IV 1:01:05
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Advanced-Level Problems On Motion In One And Two Dimensions V 1:07:34
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Advanced-Level Problems On Motion In One And Two Dimensions VI 1:08:18
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Problems On Motion In One And Two Dimensions (CE) 1:21:28
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