Question

A body travels along a straight level road. For the first half time, its speed \[{v_1}\] ​and for the second half time ${v_2}$​. What is the average speed? What would the average speed be if the first half distance is covered with speed ${v_1}$​ and the second half distance is covered with speed ${v_2}$​?

Answer 1

Hint: Average Speed is important to understand the concept of how and the rate at which a journey takes place. Throughout a journey, the speed may vary from time to time. In that case, finding the average speed becomes important to have an estimate of the rate at which the journey is completed.

Formula used: The average speed equation is articulated as:
${V_{avg}} = \dfrac{{{D_{total}}}}{{{T_{total}}}}$

Complete Answer:
Given that a body is traveling in a straight level road and during its first half of time it’s moving with velocity ${v_1}$and during the second half of time it is traveling with velocity ${v_2}$.
Let “2x” be the total distance covered.
Let ${t_1}$be the time taken to travel first half of distance and is given by:
${t_1} = \dfrac{x}{{{v_1}}}$
Similarly, ${t_2}$be the time taken to cover second half of distance:
${t_2} = \dfrac{x}{{{v_2}}}$
Then, the total time taken to cover entire distance is,
$T = {t_1} + {t_2}$
$ \Rightarrow T = \dfrac{x}{{{v_1}}} + \dfrac{x}{{{v_2}}}$
$ \Rightarrow T = \dfrac{{x({v_2} + {v_1})}}{{{v_1}{v_2}}}$
Average speed is defined as, the average speed is defined as total distance travelled by the object in a particular time interval.
Therefore, average speed is given by:
$ \Rightarrow {V_{avg}} = \dfrac{{{D_{total}}}}{{{T_{total}}}} = \dfrac{{2x}}{T}$
$ \Rightarrow {V_{avg}} = \dfrac{{2x}}{{\dfrac{{x({v_1} + {v_2})}}{{{v_1}{v_2}}}}}$
$ \Rightarrow {V_{avg}} = \dfrac{{2{v_1}{v_2}}}{{({v_1} + {v_2})}}$
Hence, the average speed of the body if the first half distance covered with speed ${v_1}$​ and the second half distance is covered with speed ${v_2}$ is $\dfrac{{2{v_1}{v_2}}}{{({v_1} + {v_2})}}$.

Note: The average speed is a scalar quantity. It is represented by the magnitude and does not have direction. The formula for average speed is found by considering the ratio of the total distance traveled by the body to the time taken to cover that distance.