Question

A car travels the first third of a distance with a speed of 10 kmph, the second third at 20 kmph and the last third at 60 kmph. What is its mean speed over the entire distance?

Answer 1

Hint: This could be solved by breaking the total distance into three parts and applying the basic formula of mean speed. The average speed is the distance per time ratio.

Formula used: Here, we will use the basic formula of speed, distance and time:
 $ {{\text{V}}_{{\text{avg}}}}{\text{ = }}\dfrac{{\text{D}}}{{\text{T}}} $
Here, $ {{\text{V}}_{{\text{avg}}}} $ is the mean speed of the car
 $ {\text{D}} $ is the total distance travel
 $ T $ is the travel time.

Complete Answer:
We will start by considering the total distance to be $ {\text{d}} $ ,
For the first third of the distance:
 $ {{\text{v}}_{\text{1}}}{\text{ = 10kmph}} $
 $ {{\text{t}}_{\text{1}}}{\text{ = }}\dfrac{{{\text{d/3}}}}{{{{\text{v}}_{\text{1}}}}} $
Similarly, for the second third of the distance:
 $ {{\text{v}}_{\text{2}}}{\text{ = 20kmph}} $
 $ {{\text{t}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{d/3}}}}{{{{\text{v}}_{\text{2}}}}} $ $ \begin{gathered}
  {{\text{v}}_{\text{1}}}{\text{ = 10kmph}} \\
  {{\text{t}}_{\text{1}}}{\text{ = }}\dfrac{{{\raise0.7ex\hbox{ $ {\text{d}} $ } \!\mathord{\left/
 {\vphantom {{\text{d}} {\text{3}}}}\right.}
\!\lower0.7ex\hbox{ $ {\text{3}} $ }}}}{{{{\text{v}}_{\text{1}}}}} \\
\end{gathered} $
And also, with $ {{\text{v}}_{\text{3}}}{\text{ = 60kmph}} $ we can write the same formula.
Now the mean speed over distance:
 $ {{\text{v}}_{{\text{avg}}}}{\text{ = }}\dfrac{{{\text{s/3 + s/3 + s/3}}}}{{{{\text{t}}_{\text{1}}}{\text{ + }}{{\text{t}}_{\text{2}}}{\text{ + }}{{\text{t}}_{\text{3}}}}} $
 $ \Rightarrow {{\text{v}}_{{\text{avg}}}}{\text{ = }}\dfrac{{\text{3}}}{{{\text{1/}}{{\text{v}}_{\text{1}}}{\text{ + 1/}}{{\text{v}}_{\text{2}}}{\text{ + 1/}}{{\text{v}}_{\text{3}}}}} $
On putting the above values,
 $ \Rightarrow {{\text{v}}_{{\text{avg}}}}{\text{ = }}\dfrac{{\text{3}}}{{{\text{1/10 + 1/20 + 1/60}}}} $
 $ \Rightarrow {{\text{v}}_{{\text{avg}}}}{\text{ = 18kmph}} $
Thus, the mean speed over the entire distance is 18kmph.

Additional Information:
Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance.
The first scientist to measure speed as distance over time was Galileo.

Note:
It should always be kept in mind that there is a difference between speed and velocity. Just as distance and displacement have distinctly different meanings, same is the situation between speed and velocity. Velocity is a vector quantity that refers to the rate at which an object changes its position whereas speed is a scalar quantity that refers to how fast an object is moving. Velocity gives us a sense of direction whereas speed does not give any sense of direction.