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Uniform Motion - Introduction
Mechanics, the oldest physical science, is the study of motion of objects. It is applied in calculation of the path of an artillery shell, a space probe sent from earth to Mars, etc.
Concept of a Point Object
A real object can rotate as it moves. For example, a cricket ball may be spinning while it is moving, as a whole, in a trajectory.
Rest and Motion
Rest and motion are relative terms. A person sitting in a moving car is at rest with respect to his or her fellow passengers, but is in motion with respect to other objects or people on the road. Moreover, it is not possible to define 'absolute rest' or 'absolute motion'. Absolute rest is the complete absence of motion which is not possible to visualise because all the objects in this universe - electrons in an atom, molecules in solids or gases, solar systems in galaxies etc., are in motion.
Motion in One, Two and Three Dimensions
A body is said to be in motion if its position changes with respect to its surrounding. In order to completely describe the motion of such objects, we need to specify its position.
Motion in a Straight Line
During motion in a straight line, the point object occupies a definite position on the path at each instant. Therefore, to describe the motion, one should specify the length of the path covered by the point object and the instant of time.
Position-Time Graph for Non-Uniform Motion
An object is said to be in non-uniform motion if it undergoes equal displacement in unequal intervals of time, however small these intervals may be.
Graphical Representation of Uniform Motion
The rate of change of position of a particle in a particular direction gives the velocity of the particle. It may also be defined as the time rate of change of displacement of a particle. Velocity is a vector quantity.
Uniformly Accelerated Motion - Introduction
Most of the motion that we come across in daily life is non-uniform motion. Moving objects are either 'speeding up' or 'slowing down'. In non-uniform motion, the velocity of the moving object changes, as a result of which the object is said to have an acceleration.
Uniformly Accelerated Motion - Introduction
The acceleration of a moving point is the rate of change of its velocity. Note that the acceleration of a moving object is a vector, as it has both magnitude and direction.
Velocity-Time Graphs of Accelerated Motion
The velocity-time graph passes through the origin and is inclined to the time-axis such that the angle of inclination is greater than 0o and less than 90o.
Relations for Uniformly Accelerated Motion (Graphical Treatment)
The velocity-time graph for uniformly accelerated rectilinear motion (motion along a straight line).
Position-Time Relations using Position-Time Graphs
In this section, Let us understand how the position changes with time when the velocity changes uniformly with time.
Distance as Area under Velocity-Time Graph
The velocity-time graph for uniformly accelerated motion of a particle is illustrated in the figure. Any two points A, B are chosen on the velocity-time graph from which perpendiculars AC, BD, AE, BF are dropped on the time and velocity axes respectively. The coordinates of the points A and B are [t, v(t)] and [tl, v(tl)] respectively.
Velocity-Time Relations using Calculus
The velocity-time relationship which has been derived graphically earlier, can also be obtained with the help of calculus.
Velocity-Distance Relation
Integrating the above equation and applying the limits which are at s = 0, initial velocity = u after a distance s has been covered, the final velocity = v.
Vectors - Introduction
A change of position of a particle is called displacement. If a particle moves from a position A to a position B.
Definitions
Collinear vectors are those vectors that act either along the same line or along parallel lines. These vectors may act either in the same direction or in opposite directions.
Unit Vector
It is a vector having unit magnitude. It is used to denote the direction of a given vector.
Fixed Vector
Fixed vector is that vector whose initial point or tail is fixed. It is also known as localised vector.
Displacement Vector
Displacement vector is a vector which gives the position of a point with reference to a point other than the origin of the coordinate system.
Multiplication of Vectors by Real Numbers, i.e., Scalar Multiple of a Vector
The multiplication of a vector by a real number assumes a lot of significance in such statements as - velocity of car B is double the velocity of car Al.
Equality of Vectors
Two vectors are said to be equal if they have the same magnitude and direction.
The Zero Vector and its Properties
Zero vector or null vector is a vector which has zero magnitude and an arbitrary direction.
Addition or Composition of Vectors
Scalars can be added algebraically. However, vectors do not obey the ordinary laws of algebra. This is because vectors possess both magnitude and direction. Vectors are added geometrically. The process of adding two or more vectors is known as addition or composition of vectors. When two or more vectors are added, the result is a single vector called the resultant vector.
Parallelogram Law of Vectors
The above example can be stated in the following way as the law of parallelogram of vectors - If two vectors, acting simultaneously at a point, can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then the resultant is represented completely, both in magnitude and direction by the diagonal of the parallelogram passing through the point.
Law of Polygon of Vectors
If a number of vectors are represented by the sides of a closed polygon taken in order, then, their resultant is zero.
Resolution and Components of Vectors in a Plane
The process of splitting a vector is called resolution of a vector. In simpler language it would mean, determining the effect of a vector in a particular direction. This is explained with the help of an example later on. The parts of the vector obtained after splitting the vector are known as the components of the vector.
Dot Product of Vectors and the Resolution of a Vector
It was mentioned earlier that, displacement vectors are added to displacement vectors, or velocity vectors are added to velocity vectors. Just as it is meaningless to add scalar quantities of different kinds, such as mass and temperature, so also it is meaningless to add vector quantities of different kinds, such as displacement and electric field strength.
Relative Velocity in the Case of Two Velocities Inclined to Each Other
When both the cars are moving in the same direction, i.e., q = 00 degrees.
Cross Product or Vector Product of Two Vectors
Imagine rotating a right handed screw whose axis is perpendicular to the plane formed by a and b so as to twist it from a to b through the angle p between them.
Motion in a Plane - Introduction
In this chapter, we will consider motion in two dimensions taken to be the X-Y plane, for convenience.
Case of Uniform Velocity
A particle is said to move with uniform velocity if it undergoes equal displacements in equal intervals of time, however small these intervals may be.
Case of Uniform Acceleration
A particle is said to move with uniform acceleration if its velocity changes by equal amounts in equal intervals of time, however small these intervals may be.
Expression for Velocity Vector
The curved line in the figure represents the trajectory of a particle. Let the particle be at P at time t.
Projectile Motion
Projectile motion is an example of curved motion with constant acceleration. This is the two dimensional motion of a particle thrown obliquely into the air. The ideal motion of a cricket ball, a golf ball or a bullet is an example of projectile motion. We assume that the effect air could have on their motion is negligible.
Oblique Projectile
v Cosq along X-axis which is constant, since, the force of gravity in the horizontal direction is zero.
Maximum Height
Maximum height is denoted by the letter hmax or H. It is also known as the vertical range. It is the maximum height to which a projectile rises above the horizontal plane of projection.
Uniform Circular Motion
The revolution of moon around the Earth and the revolution of an artificial satellite in a circular orbit round the Earth are examples of circular motion.
Summary
Definitions of displacement, uniform velocity, variable velocity, average velocity, speed and instantaneous velocity with examples.
