physics

 
A body
A certain amount of matter limited in all directions and consequently having a finite size, shape and occupying some definite space is called a body.
Particle
A particle is defined as a portion of matter infinitesimally small in size so that for the purpose of investigation, the distance between its different parts may be neglected. Thus, a particle has only a definite position, but no dimension. In the problems we are going to discuss, we will consider a body to be a particle for the sake of simplicity.
MOTION IN ONE DIMENSION
Motion
The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point on frame. So we can define motion as follows:-
An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference).
Caution :  The moving object is either a particle, a point object (such as an electron) or an object that moves like a particle. A body is said to be moving like if every portion of it moves in the same direction and at the same rate.
Motion in One Dimension
When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.
To understand the essential concepts of one dimensional motion we have to go through some basic definitions.
Frame of reference
One can see the platform from a running train, and it seems that all the objects placed on platform are continuously changing their position. But one, who is on platform, concludes that the objects on the platform are at rest. It means if we will take the trains are reference frame the objects are not stationary and taking reference frame as platform the objects are stationary. So the study of motion is a combined property of the object under study and the observer. Hence there is a need to define a frame of reference under which we have to study the motion of an object.
Definition
A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer's requirement. The essential requirement for a frame of reference is that, it should be rigid.
Position of an object 
The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence X, Y, Z is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that pointIntroduction to Motion in One Dimension Position vector
It describes the instantaneous position of a particle with respect to the chosen frame of reference. It is a vector joining the origin to the particle. If at any time, (x, y, z) be the Cartesian coordinates of the particle then its position vector is given by vector = xi + yj + zk.
In one-dimensional motion: vector = xi,     y = z = 0   (along x-axis)
In two-dimensional motion: vector = xi + yj                 (in x-y plane z = 0)

        In the figure above, the position of a point P is specified and vector OP is called the position vector.

Caution:    Motion of a body cannot be correctly identified unless one knows the position of body as specified by a fixed frame of reference.

Displacement
        Consider a case in which the position of an object changes with time. Suppose at certain instant 't' the position of an object is x₁ along the x axis and some other instant 'T' the position is x₁ then the displacement Δx is defined as

Δx = x₂ - x₁

        It can be seen in the figure above where x₁ and x₂ are instantaneous position of the object at that time.
                                   
        Now consider the motion of a point A with respect to a reference point O. The motion of point A makes its radius vector vary in the general case both in magnitude and in direction as shown in figure above. Suppose the point A travels from point 1 to point 2 in the time interval Δt. It is seen from the figure that the displacement vector Δ of the point A represents the increment of vector in time Δt:Introduction to Motion in One Dimension

Difference between distance and displacement
To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O, as shown in the figure below, to height h.

After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It's because distance is a scalar quantity but displacement is a vector quantity.
Uniform and Non Uniform Motion
Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.
The motion of an object may be uniform or non-uniform depending upon its speed. In case of uniform motion the speed is constant, whereas in the non-uniform motion, the speed is variable.
       In uniform motion in one dimension the velocity (v) is mathematically defined as

v = (x₂ - x₁)/(T-t)     ...... (1)

        where x₁ and x₂ are instantaneous displacement as shown in figure above at time 't' and 'T' respectively.
Graphical representation of the uniform motion
        Form the equation (1) we have the following equation

x₂ = x₁ + v(T - t)

        where v is constant. Take t = 0, the equation becomes x₂ = x₁ + vT, from this equation it follows that the graph of position of object 'x₂' against 'T' is a straight line, cutting off x₁ on the position axis where x₁ is the distance of the particle from the origin at time t = 0.
        v = slope of the graph which is constant

Note:  If the graph is not a straight line, it will represent non-uniform motion.Introduction to Motion in One Dimension

Velocity Vector in Non Uniform Motion
        In any non-uniform motion, we can define an average velocity over a time interval. Average velocity is the ratio of the displacement Δx (that occurs during a particle time interval Δt) to that interval of time i.e.

        Now refer to the example, related to figure 2.3, the ratio of Δ/Δt is called the average velocity <> during the time interval Δt. The direction of the vector <> coincides with that of Δ. Average velocity is also a vector quantity.

Note:  The ratio of total distance traveled and time taken during the motion is called average speed. Average speed is a scalar quantity.

If at any time t₁ position vector of the particle is1 and at time 2 position vector is 2 then for this interval
Instantaneous velocity
Instantaneous velocity is defined as the rate of change of displacement.
                                                  
Illustration:
This question contains statement-1 (Assertion) and Statement-2 (Reason). Question has 4 choices (A), (B), (C) and (D) out of which only one is correct.
Statement-1
A bus moving due north take a turn and starts moving towards east with same speed. There will be no change in the velocity of the bus.
Statement-2
Velocity is a vector quantity.
(A)  Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1.
(B)  Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for statement-1.
(C)  Statement-1 is true, Statement-2 is false.
(D)  Statement-1 is false, Statement-2 is true.
Solution (D)
     This is so because bus is changing its direction of motion.
Illustration:
      A man started running form origin and went up to (2, 0) and returned back to (-1, 0) as shown in figure 2.7. In this process total time taken by man is 2 seconds. Find the average velocity and average speed.

Introduction to Motion in One Dimension

Solution:
        The man is displaced form origin to (-1, 0)
        Hence displacement, s =
        So average velocity  = Displacement/total time = /2 = (-1/2)m/sec.
        where as, since the total distance traveled by man
                                = (0, 0) to (2, 0)+(2, 0) to (0, 0)+(0, 0) to (-1, 0)
                                = 2 + 2 + 1 = 5 m
        Hence average speed
                                = (Total distance)/(Total time)=5/2 m/sec.
Velocity
        The velocity at any instant is obtained from the average velocity shrinking the time interval closer to zero. As Δt tends to zero, the average velocity approaches a limiting value, which is the velocity at that instant, called instantaneous velocity, which is a vector quantity, mathematically we can define it as

        The magnitude v of the instantaneous velocity is called the speed and is simple the absolute value of
        In the example related with figure given below, the instantaneous velocity is

                                   
        Hence instantaneous velocity is the rate at which a particle's position is changing with respect to time at a given instant. The velocity of a particle at any instant is the slope (tangent) of its position curve at the point representing that instant of time, as shown in figure above.
Speed
Speed is defined as rate of change of distance with time.
In any interval of time, average speed is defined as
<speed> = (total distance)/(total time taken) = Δs/Δt. As Δs > |Δ|, hence <speed> > <velocity>
Think : (i) Can a body have a constant speed and still have a varying velocity?
                (ii) Can a body have a co