Let one-way distance = x km
Time is taken in forwarding trip at a speed of 20 km/h = \(\frac { Distance }{ Speed } =\frac { X }{ 20 } h\)
Time is taken in return trip at a speed of 30 km/h = \(\frac { X }{ 30 } h\)
Total time for the whole trip = \(\frac { Distance }{ Speed } =\frac { x }{ 20 } +\frac { x }{ 30 } =\frac { 3x+2x }{ 60 } =\frac { 5x }{ 60 } h\)
Total distance covered = x + x = 2 x  km
Average speed =\(\frac { Total\quad Distance }{ Total\quad time } =\frac { 2x }{ 5x/60 } +\frac { 2xX60 }{ 5x/60 } \) 
 = 24 km \(h^{ -1 }\) 

(a) For motion from A to B:
Distance covered = 300 m
Displacement covered = 300 m
Time taken = 2 minutes 30 seconds =2 x 60 + 30 = 150 s
Average speed = \(\frac { Distance\quad covered }{ Time\quad taken } =\frac { 300m }{ 150\quad s } =2\ ms^{ -1 }\)
Average velocity = \(\frac { Displacement }{ Time\quad taken } =\frac { 300m }{ 150\quad s } =2\ ms^{ -1 }\)
(b) For motion from A to B C:
Distance covered = 300 + 100 = 450 m
Displacement covered = AB - CB = 300 - 100 = 200 m
Time taken = 2.50 + 1.00 = 3.50 min = 210 s
Average speed=\(\frac { Displacement\quad covered }{ Time\quad taken } =\frac { 400m }{ 210\quad s } =1.90\ ms^{ -1 }\)
Average velocity= \(\frac { Displacement\quad covered }{ Time\quad taken } =\frac { 200m }{ 210\quad s } =0.952\ ms^{ -1 }\) 

Time taken = 2 min 20 s = 2 x 60 + 20 = 140 s
Radius,r = 100 m
In 140 s, the athlete will complete three and a half round
Distance covered = \(2\pi \) x 3.5
= 2 x \(\frac { 22 }{ 7 } \) x 100 x 3.5 = 2200 m.
At the end of his motion, the athlete will be in the diametrically opposite position
\(\therefore \)  Displacement = diameter = 200 m

Third equation of motion. Let a body start with initial velocity u and after covering distance s under uniform acceleration a, its velocity v in t seconds.Then 
Average velocity = \(\frac { u+v }{ 2 } \)
So the distance covered in time t is given by
s = Average velocity xTime = \(\frac { u+v }{ 2 } Xt\)
v + u = \(\frac { 2s }{ t } \) 
Using the first equation of motion : v = u+at or v - u = at
Multiplying equations (i) and (ii) ,we get 
(v+u) (v-u) = \(\frac { 2s }{ t } \) x at or  
\(v^{ 2 }-u^{ 2 }=2as\) 

The second equation of motion. Suppose a body starts with initial velocity u and due to uniform acceleration a, its final velocity becomes v after t. Then
Average velocity = \(\frac { Intial\quad velocity+Final\quad velocity }{ 2 } =\frac { u+v }{ 2 } \)
So, the distance covered by the body in time t is
s = Average velocity x Time
=\(\frac { u+v }{ 2 } \) x t=\(=\frac { u+(u+at) }{ 2 } Xt\)   \([\therefore v=u+at]\)
\(=\frac { 2ut+at^{ 2 } }{ 2 } \)
or \(s=ut+\frac { 1 }{ 2 } at^{ 2 }\) 

Acceleration. In non-uniform motion, the velocity of a body changes with time It has different velocities at different instants of time and at different points of its path In such a situation we define a physical quantity called acceleration which is a measure of the change in the velocity of a body per unit time.
Acceleration is defined as the rate of change of velocity.If the velocity of a body changes from u to v in time t, then
Acceleration=\(\frac { Change\quad in\quad velocity }{ Time\quad taken } =\frac { Final\quad velocity-Initial\quad velocity }{ Time\quad taken } \)
or \(a=\frac { v-u }{ t } \)
As acceleration has both magnitude and direction, it is a vector quantity.

Average speed. When the speed of the body varies with time  we need to define its average speed.
Average speed is the total distance travelled by a body, divided by the total time taken to cover that distance Thus,
Average speed=\(\frac { Total\quad distance\quad travelled }{ Total\quad time\quad taken } \)
For example, if a car travels a distance of 10 km in 2 hours, then its
Average speed=\(\frac { 100\quad km }{ 2\quad hour } \) = 50 km per hour.

Different types of Motions:
1.Translatory Motion When a body moves as a whole along a straight or curved path, it is said to be in translatory motion.It is again of two types:
(i) Rectilinear motion Here a body moves as a whole along a straight path.For example a train moving on straight rails has translatory rectilinear motion.
(ii) Curvilinear motion.Here a body moves as a whole along a curved path For example motion of a bicycle taking a turn along a curved path.
2.Rotatory motion.When a body rotates about a fixed point or axis , it has rotatory motion For example motion of a flywheel about a shaft.
3.Vibratory or oscillatory motion. When a body moves to and from about a mean position again and again it has vibratory or oscillatory motion For example the motion of the pendulum of a wall clock. 
4.Complex motion. Sometimes ,the motion of a body may be a combination of more than one types of motion For example a ball rolling down an inclined plane has botth translatoruy and rotatory motions.

a = 8 \(ms^{ -2 }\),v = 32 \(ms^{ -1 }\) 
(ii) t = \(2\sqrt { 2\quad s } \)

\(v_{ av }=\frac { s_{ 1 }+s_{ 2 } }{ t_{ 1 }+t_{ 2 } } =\frac { s }{ \frac { s_{ 1 } }{ v_{ 2 } } +\frac { s_{ 2 } }{ v_{ 2 } } } \)
\(40=\frac { 100 }{ \frac { 30 }{ 30 } +\frac { 70 }{ v_{ 2 } } } \)
\(=1+\frac { 70 }{ v_{ 2 } } =\frac { 100 }{ 40 } =\frac { 5 }{ 2 } \)
\(\frac { 7 }{ v_{ 2 } } =\frac { 5 }{ 2 } -1=\frac { 3 }{ 2 } \)
\(v_{ 2 }=\frac { 7X2 }{ 3 } =\frac { 14 }{ 3 } =4.67\quad km\quad h^{ -1 }\)