When food packets are to be dropped to marooned people during flood relief operations food packets are dropped from aeroplanes. How much is to be the distance of dropping the food packets from the people?
A boy runs behind a bus to catch it. With what velocity should he run to catch the bus? When the fuel of a rocket is exhausted at a certain height does it stop there only?
To know the answers of such questions read the following.
To describe the motion of an object, one should know its initial position i.e, at time
t = 0 and its position at different times. From these positions at different times one can determine the displacement, velocity and acceleration of the object. This gives a complete description of motion of the object. While describing the motion one need not consider the effect of forces on the object. Such a study is called kinematics.
The motion of an object is not absolute because to find the position we have to consider a coordinate system called reference frame.
Different persons may consider different coordinate systems to describe the motion and may describe the motion differently. So, motion is not absolute. With reference to a particular reference frame one has to describe the motion of an object.
Displacement: To find the displacement of an object we have to find its initial position i.e, its position (xi) at time t = 0 and its position at time t called final position (xf) along a straight line taken to be X-axis.
Then xf - xi gives the displacement (s) of the object
What ever may be the path followed by the object to travel from xi to xf,
the displacement of the object is taken along the straight line joining the initial and final positions. It is a vector with its tail at initial position and head at the final position. 
Distance travelled: To travel from initial to final positions the object may follow different paths.
e.g.: To go from your house to you may not travel along a straight line. You will travel along the roads available.
Total length of the path along which an object travels from initial to final positions gives the distance travelled. AA1B, AA2B, AA3B are the different distances travelled by the object from A to B.
If the body follows the shortest path i.e., along a straight line joining the initial and final positions distance travelled becomes equal to the displacement of the object.
The velocity of an object at any instant of time is its instantaneous velocity. It is equal to the rate of change of displacement of the object.
Where ∆s is change in displacement in a time interval ∆t. In the limit ∆t tends to zero 
tends to 
which is the derivative of s with respect to time. One can find the instantaneous velocity of a car by seeing its speedometer.
When an object travels from A to B along a straight line path in time interval t1 and from B to C along another straight line path in time interval t2, its velocity during the time interval (t1 + t2) is the average velocity of the object. It is given by
To determine the speed of the object during the same time interval, the total distance
travelled along the path ABC is AB + BC and then
e.g.: If an object travels 3 m towards east in 1 second and then 4 m towards north in another 1 second, its displacement during 2 seconds is 5 m and its
Velocity is a vector but speed is a scalar.
Uniform and non-uniform Velocities: When an object makes equal displacements in equal time intervals however small these may be, then the body is said to have uniform velocity. From this, it can be seen that the velocity is same at all times in a given time interval for a body to have uniform velocity. This is possible when the object travels along a straight line. In such a case, instantaneous velocity becomes equal to average velocity.
If the velocity of the object changes with time then its velocity is non-uniform. The change of velocity may be in its magnitude or direction or both. Such an object will have acceleration.
Acceleration is rate of change of velocity.
Instantaneous acceleration is the acceleration of an object at a particular instant of time which is given by
When a body having initial velocity u moves with uniform acceleration a its final velocity becomes 
after time t and displacement s, its kinematical equations are
= u + at, s = ut + 
at2 and 2 - u2 = 2as.
The final position xf can be obtained by substituting s = xf - xi in s = ut + at2
xf - xi = ut + at2 xi = xf + ut + at2
The displacement during nth second is sn = u + a(n - )
Note: (1) As the object moves along a straight line distance travelled is equal to its displacement. (2) Kinematical equations are applicable only for uniformly accelerated objects travelling along a straight line.
x-t graph: A graph plotted between position and time is called x-t graph. Nature of x-t graph can be found from xf = xi + ut + at2. When xf is taken as x.
i) If an object is at rest. The nature of x-t graph can be obtained by putting u = 0, a = 0 and xf = x in xf = xi + ut + at2 This gives xf = xi and the graph is a straight line parallel to X - axis
ii) If an object moves with a constant velocity , substituting
u = , a = 0 and xf = x we get x = xi = t The nature of x - t graph is a straight line with Y- intercept xi and slope equal to its velocity .
iii) If an object has uniform acceleration, the nature of x-t graph is a parabola. The parabola is concave upwards for positive acceleration and convex upwards for negative acceleration.
Note 1) s-t graphs are similar to x-t graph in nature. The nature of s-t graph is found from
s = ut + at2.
2) Slope of these graphs gives the instantaneous velocity at that instant of time.
3) If the graph is a curve the slope of the tangent drawn at a particular instant of time gives the instantaneous velocity.
- t graph: Taking velocity of an object along Y-axis and time along X-axis, a graph plotted is - t graph. Its nature can be found from the kinematical equation = u + at.
i) - t graph of an object at rest is X-axis.
ii) - t graph of an object moving with uniform velocity is a straight line parallel to X-axis.
iii) - t graph of an object moving with uniform acceleration is a straight line with Y - intercept equal to initial velocity and slope equal to acceleration. It is inclined
upwards for positive acceleration and inclined downwards for negative acceleration.
Note: i) Slope of - t graph is equal to acceleration.
ii) Area under - t graph i.e., area between - t graph and time-axis is equal to the displacement.
Motion under gravity
When a body is projected vertically upwards or falling freely its acceleration is called acceleration due to gravity denoted by g. This can be taken to be constant for small heights from the earth.
A body projected vertically upwards has initial velocity u, acceleration - g. So, its kinematical equations are = u - gt, y = ut - gt2, 2 - u2 = -2 gy, where y is its vertical displacement.
A body falling freely through a height h has initial velocity u = 0, acceleration a = +g. Its kinematical equations are = gt, h = gt2, 2 = 2gh.
Note: 1) When a body is projected vertically upwards while it passes through a point upwards and then downwards, the magnitude of the velocity is same. Its directions are opposite.
2) When a body is projected vertically upwards from a height h its kinematical equations are obtained by substituting initial velocity u = u, acceleration a = -g and displacement s = -h in kinematical equations. We get
= u - gt; -h = ut - gt2 and 2 = u2 + 2gh.
3) When a body is thrown vertically downwards with an initial velocity u, its acceleration a = +g and equations of motion are = u + gt, h = ut + gt2 and 2 - u2 = 2gh.
4) The final velocity on reaching the ground will be the same if a body is thrown vertically upwards or downwards from the same height h with the same magnitude of initial velocity.
Projectile Motion:
A projectile is an object thrown at an angle other than 90° with the horizontal. It does not follow a straight line path. Its motion is two dimensional motion which is plane motion. The path or trajectory of a projectile is a parabola.
A projectile may be oblique or horizontal.
Oblique projectile:
Time of flight is the time taken by a projectile to reach the horizontal level of projection.
Maximum height is the height of a place at which the vertical component of velocity becomes zero.
Horizontal range is the maximum horizontal displacement of a projectile when it reaches the horizontal level of projection.
