Motion is a combined property of the object under study and observer.

If the position of the object under study changes with time, as seen by the observer, the object is said to be in motion from the frame of reference of the observer.

If position of the object does not change with time, as seen by observer, object is said to be at rest from the frame of reference of the observer.

Rest and motion of an object under study depend on the frame of reference of the observer. For eg. A book kept on a table may be at rest for all students sitting in the class. But the same book will be in motion, as seen by an observer on a moving bus. Thus absolute rest and absolute motion are meaningless.

In most cases, if attributes of motion of an object are given without specifying the frame of the observer, it is to be assumed that the object under consideration is being observed by an observer who is at rest with respect to the earth.

For a particle moving along a straight line, position of the particle can be specified with only one coordinate. A coordinate system is chosen by choosing some reference point as the origin. The origin is assigned the number zero. Most situations can be analysed by setting up an appropriate coordinate system. In order to do so, following are the essential requirements:

`=>` Choice of origin
`=>` Choice of coordinate axis

Choice of positive direction of axis. All these parameters constitute a reference frame. In any physics problem, the reference frame must be specified.

In the figure below, point `O` is the chosen origin, `X` -axis is the chosen coordinate axis and rightward direction is chosen as the positive direction.

Similarly, if the motion of a particle is `2` - dimensional or `3` - dimensional, the coordinate axes will comprise of `x, y` and `z` axes and position will include `x, y`, and `z` coordinates.

Displacement is a vector quantity. It is the change in position vecto `r`. Distance is the total length of the actual path covered. Distance is a scalar quantity.

Suppose a particle travels from point `A` to point `B` as shown in the fig below along a zig - zag path, in a finite time interval.

Coordinates of `A` are `(x_1, y_1)` and that of `B` are `(x_2,y_2)` . Position vector of `A` is `vec r_(A)=x_1 hat i +y_1 hat j`, position vector of `B` is `vec r_B=x_2hat i+y_2 hat j` . Distance will be equal to the total length of the actual path covered by the particle. Displacement will be `vec S = vec r_B-vec r_(A)=(x_2-x_1) hat i+ (y_2-y_1)hat j`

=>The distance covered will always be greater than or equal to the magnitude of the displacement.

=>Displacement and distance are equal in magnitude in case the particle is travelling along a straight line without change in direction.

=>`SI` unit of distance and displacement is meters.

=>In simple language, displacement can be said to be the shortest line joining the initial and final positions of a body in motion, irrespective of path followed and it is directed from initial position to final position.

=>Change in position vector is displacement and change in displacement vector is also displacement.

`text(Average speed) = text(Tolal Dislance travelled)/text(Total time taken)`,

We define Total time taken average speed of a paticle as the ratio of the total distance travelled to the total time taken.

`SI` units of speed is `m // s`

Speed of a particle at a particular instant is called instantaneous speed.

The speedometer of a vehicle indicates the instantaneous speed. The speedometer reading on a crowded city road continuously changes, indicating instantaneous speed is continuously changing.

Velocity is defined as rate of change of displacement with time. Velocity is a vector quantity. `SI` unit of velocity is `m//s`.

`text(Average velocity)= text(Total Displacement)/text(Total time)` Average velocity is defined as the ratio of the total displacement covered to the total time taken.

Just as distance is always greater than or equal to the magnitude of displacement, average speed is greater than or equal to the magnitude of average velocity. Average speed and the magnitude of average velocity are equal when particle is travelling in a straight line without change in direction.

Suppose a particle moves from position `x` at time `t` to position `x + Delta x` at time `t+Delta` Then, the average velocity of the particle over time interval `Delta t` is `(Delta x)/(Delta t)`.

Making `Delta t` infinitely small, `(Delta x)/(Delta t)` gives the velocity of the particle at instant `t` and can be written as ` v=lim_(Delta t-> 0) (Delta x)/(Delta t) =(dx)/(dt)` where `v` is the instantaneous velocity of the particle at time instant `t`.

The magnitudes of instantaneous velocity and instantaneous speed are always equal.

Motion of a body in a straight line with uniform velocity is called uniform motion .

`(ds)/(dt)=v` , but `v` is constant in uniform motion.

`:. int ds =int vdt=v int dt =vt`.

`:. s=vt`, where `s` is the displacement, `t` is the time interval, `v` is uniform velocity.

Uniform motion can also be said to be motion in which equal displacements are covered in equal intervals of time, however small the time intervals may be.

A body thrown vertically upwards or vertically downwards or dropped from a height will move in a straight vertical line.

lf air resistance is ignored, the body will be subjected to acceleration due to gravitational force exerted by the earth, which is denoted by `g`. The value of `g` on the earth is `9.8m//s^2` in the downward direction.

For small heights, the value of `g` is constant, we can use equations of uniformly accelerated motion.

We shall take upward direction as positive & down direction as negative, as our convention.


`text(Motion of a particle projected downwards from height)` `h` `text(above surface of earth :)`

Suppose a particle is projected downwards from height `h` above the surface of the earth with speed u. To find the time taken by it to strike the surface of the earth, taking upward direction as positive, `u=-u, a= - g, s = -h` , apply `s= ut+1/2 at^2` , solve the quadratic and get the positive value of `t`.

`text(Motion of a particle dropped from a height)` `h` `text(above surface of earth :)`

Solve using `v^2 = u^2+2as` and `s=ut+1/2 at^2`, taking `u = 0`, Velocity with which it strikes the surface will be `sqrt(2gh)` and the time it will take to strike the surface will be `sqrt((2h)/g)`