Problems related to boats and streams are
different in the computation of relative speed from those of trains/cars. .
When a boat is moving in the same direction
as the water current/stream, the boat is said to be moving WITH THE STREAM OR
CURRENT. .
When a boat is moving in a direction
opposite to that of the water current/stream, it is said to be moving AGAINST
THE STREAM OR CURRENT. .
SPEED OF THE BOAT IN STILL WATER is the
speed of the boat when it travels in water that is not moving. .
When the boat is moving upstream, the speed
of the water opposes (and hence reduces) the speed of the boat. .
When the boat is moving downstream, the
speed of the water aids (and thus adds to) the speed of the boat. Thus, we have
.
Speed of the boat against stream = Speed of
the boat in still water - Speed of the stream. .
Speed of the boat with the stream = Speed of
the boat in still water + Speed of the stream. .
These two speeds, the speed of the boat
against the stream and the speed of the boat with the stream, are RELATIVE
SPEEDS. .
In problems, instead of a boat, it may be
any other moving body, the approach is exactly the same. .
Circular Tracks
When two people are running around a
circular track starting at the same time from the same point, we would be
interested in finding out the time taken by them to meet for the first time
anywhere on the track or to meet for the first time at the starting point. .
This can be done when the two people are
running in the same direction or in the opposite direction. .
Similarly, we can find the time taken for
THREE people to meet each other to meet for the first time anywhere on the
track or at the starting point. .
Let the length of the circular track be L.
Let three people be A, B and C run around the circular track with speeds of a,
b and c respectively where a > b > c. They start running from the same
point at the same time.
Then the following table gives complete
details for finding out the time required for each of the various meetings as
discussed above. .
|
Time taken / Situation
|
Two people A and B running in
opposite directions
|
Two people A and B running in
the same direction
|
Three people A, B and C running
in the same direction
|
|
To meet for the first time anywhere on the
track
|
|
|
LCM of {L/(a - b), L/(b - c)}
|
|
To meet for the first time at the starting
point
|
LCM of {L/a, L/b}
|
LCM of {L/a, L/b}
|
LCM of {L/a, L/b, L/c}
|
Also, when two people are running around a
circular track starting from the same point at the same time, then every time
the two people meet, the faster person covers one full round more than the
slower person and vice-versa. .
Some solved example
example.1
Walking at 5 kmph a student reaches his
school from his house 15 minutes early and walking at 3 kmph he is late by 9
minutes. What is the distance between his school and his house? .
Solution
Let the distance between school and house be
x km. .
&⇒ 10x = 30
&⇒ x = 3 km. .
Example.2
In a 1000 m race, A beats B by 100 m and C
by 180 m. In a 1800 m race, by how many meters does B beat C? .
Solution
By the time A covers 1000 m, B covers 1000 -
100 = 900 m and C covers 1000 - 180 = 820 m. .
By the time, B covers 1800 m, the distance C
covers = 
x 820 =
1640
So, B beats C by 1800 - 1640 = 160 m. .
example.3
A man can swim at 6 kmph downstream and 4
kmph upstream. Find the speed of the man in still water and also the speed of
the stream. .
Solution
Let the
speed of the man in still water be x kmph and speed of the stream be y kmph
x + y = 6 ------- (1) x - y = 4 -----------
(2)
Solving the equations we get x = 5 and y = 1
example.4
A boat travelled from A to B and back to A
from B in 5 hours. If the speed of the boat in still water and speed of the
stream be 7x5 kmph and lx5 kmph respectively, then what is the distance between
A and B? .
Solution
Let
distance between A and B be x km
Speed upstream = 7 x 5 – 1 x 5 = 6 kmph
Speed downstream = 7 x 5 + 1 x 5 = 9 kmph
= 5
&⇒ x = 18
example.5
The distance between two persons is 800 m.
If they start moving towards each other simultaneously at 10 m/s and 15 m/s, in
how much time do they meet? .
Solution
Distance = 800 m
Relative speed = (10 + 15) = 25 m/s
Time taken to meet = 
= 
= 32
sec
