Question

A body travelling along a straight line traversed one half of the total distance was covered with a velocity $V_0$. The remaining part of the distance was covered with a velocity $V_1$, for half of time and with velocity $V_2$ for the other half of time. Find the mean velocity averaged over the whole time of motion.

Answer 1

Kinematics is the study of motion of the particular bodies without considering the reason of motion. First calculate the distance of the first half then calculate the time taken to cover the second half. Then calculate the total time and using the above data find the mean velocity.

Formula used:
$speed = \dfrac{{dis\tan ce}}{{time}}$

Complete step by step answer:
The study of motion of the bodies without considering the cause of motion is called kinematics. Distance between initial and final position is called length.
Displacement may or may not be equal to the path length travelled by an object.
Distance travelled by unit time is called speed. It is a scalar quantity.
A body is in uniform motion along a straight line when that body is moving with uniform velocity.
A particle moving along a straight-line travels an equal distance in equal intervals of time, then the particle is moving at uniform speed.

A particle moving along a straight-line travels unequal distance in equal intervals of time or equal distance in unequal time then the particle is moving with no uniform field.
Total distance = $s$
The time taken to travel half the distance is ${t_1}$
Time taken by second half is ${t_2}$
The time taken for the first half distance
${t_1} = \dfrac{{\left( {\dfrac{S}{2}} \right)}}{{{V_0}}} = \dfrac{S}{{2{V_0}}}$
Then the time taken for the second half
${t_2} = \dfrac{S}{{{V_1} + {V_2}}}$
Then the total time is $t = {t_1} + {t_2}$
$t = \dfrac{S}{{2{V_0}}} + \dfrac{S}{{{V_1} + {V_2}}}$
Then the average speed will be $\dfrac{\text{distance}}{\text{time}}$
$\text{speed} = \dfrac{S}{{\dfrac{S}{{2{V_0}}} + \dfrac{S}{{{V_1} + {V_2}}}}}$
Taking LCM on the denominator term and we get,
$\text{speed}= \dfrac{S}{{\dfrac{{S\left( {\left( {{V_1} + {V_2}} \right) + 2{V_0}} \right)}}{{2{V_0}\left( {{V_1} + {V_2}} \right)}}}}$
Taking reciprocal of the term and cancel the same term we get,

Note: Distance and displacement are two different quantities. Distance has only magnitude, whereas the displacement has both magnitude and direction. Displacement’s magnitude may or may not be equal to the path length travelled by an object.