Question

At noon, ship A starts from a point P towards a point Q and at $1.00\,pm$ ship B starts from Q towards P. If the ship A is expected to complete the voyage in $6\,hours$ and ship B is moving at a speed $\dfrac{2}{3}$that of ship A, at what time are the two ships expected to meet one another?
A.$4pm$
B.$4:30pm$
C.$3pm$
D.$2:30pm$

Answer 1

Hint: Try to find the relative speed of the ships and distance at the same time so that the time can be evaluated by the ratio of distance to the relative speed.

Complete step-by-step solution:
First, we assume that the distance between the two ships A and B is $d$.
The ship A is expected to complete the voyage in $6\,hours$.
Let the speed of the ship A is${S_1}$. Evaluate speed of the ship A.
Speed is the ratio of the distance covered to the total time taken.
${S_1} = \dfrac{d}{6}$
Let the speed of the ship B is ${S_2}$it is $\dfrac{2}{3}$that of ship A , therefore, \[{S_2} = \dfrac{2}{3}\dfrac{d}{6} = \dfrac{d}{9}\].
Since both the ships are moving towards each other, therefore, the relative speed of the two ships is the sum of their individual speeds.
Let the relative speed is $S$.
Hence, the relative speed is,
$
  S = \dfrac{d}{6} + \dfrac{d}{9} \\
  S = \dfrac{{5d}}{{18}} \\
 $
Since ship A starts at noon means at $12\,pm$ and ship B starts at $1\,pm$, therefore, first evaluate the distance covered in $1\,hour$ by ship A.
Distance is the product of speed and time.
$\dfrac{d}{6} \times 1 = \dfrac{d}{6}$
Now evaluate the distance between the ships at $1\,pm$.
\[d - \dfrac{d}{6} = \dfrac{{5d}}{6}\]
Let after $t\,hours$ the two ships expected to meet one another
We know that time is the ratio of distance to the relative speed.
Evaluate the time by substituting the value of distance and relative speed.
\[
  t = \dfrac{{\dfrac{{5d}}{6}}}{{\dfrac{{5d}}{{18}}}} \\
  t = 3\,hours \\
 \]
Hence, after $3\,hours$ from $1\,pm$ the ship will meet one another that is $1 + 3 = 4\,pm$
Therefore, option A is correct.
Note: In this type of questions don’t forget to evaluate the distance at the same time instant and relative speed is to be calculated on the basis of the direction of ships. If they are in the same direction then the relative speed will be the difference of speeds and if they are in opposite directions or towards each other then the relative speed will be the sum of speeds.