Question

An insect moves along the sides of a wall of dimensions \[12m \times 5m\]starting from one corner and reaches the diagonally opposite corner. If the insect takes \[2s\] for its motion then find the ratio of average speed to average velocity of the insect.

(A) 17:13
(B) 12:5
(C) 13:5
(D) 17:12

Answer 1

Hint We know the total dimension of the wall and it is given that the insect travels the diagonal path. This diagonal path is the shortest distance and hence can be calculated using Pythagoras theorem. Find the average velocity by dividing displacement and time and average speed by distance and time and find its ratio.

Complete Step by Step Solution Average speed of a body or particle is defined as the total distance or path covered by the body or particle in the given time period. It is the ratio of the distance travelled to the time of travel. Whereas, average velocity of a body is defined as the total displacement of the body from one point to another per unit time.
Now in our given problem , the insect is said to travel in a diagonal path from one corner to another. This is said to be the shortest distance of travel by the insect and hence by definition , becomes the displacement of the insect from one corner to another. We don’t know the distance but we can calculate using Pythagoras theorem for the triangle formed by the diagonal.
In \[\Delta ABC\],
\[ \Rightarrow A{C^2} = \sqrt {A{B^2} + B{C^2}} \], where AC is the diagonal length and AB and BC are sides of the wall
\[ \Rightarrow A{C^2} = \sqrt {{{12}^2} + {5^2}} \]
\[ \Rightarrow AC = 13m\]
Hence, we know that the displacement of the insect is 13m. Now , using the definition of average velocity, we can calculate by dividing the total displacement achieved and the given time period.
\[AvgVel = \dfrac{{TotalDisp}}{{Time}}\]
\[ \Rightarrow AvgVel = \dfrac{{13}}{2}\]
Now, the total distance travelled by the insect had it not travelled diagonally will be the sum of the sides of the wall, which is\[12m + 5m = 17m\]. Now, average speed is given as the ratio of total distance travelled and the time period of travel.
\[ \Rightarrow AvgSpeed = \dfrac{{TotalDist}}{{Time}}\]
\[ \Rightarrow Avg\ Speed = \dfrac{{17}}{2}\]
Now, the ratio of average speed to average velocity is given as
\[ \Rightarrow \dfrac{{AvgSpeed}}{{AvgVel}} = \dfrac{{\dfrac{{17}}{2}}}{{\dfrac{{13}}{2}}} = \dfrac{{17}}{{13}}\]

Hence, Option (a) is the right answer for the given question.

Note Average velocity of a particle or body , can be kinematically defined as the rate of change of the position of the body in the reference frame, with time as it’s function. It is an essential kinematic constraint that is keenly observed during vehicle racing.