08 Time, Speed and Distance

When a boat is moving in the same direction as the stream or water current, it is said to be moving WITH THE STREAM OR CURRENT.

When a boat is moving in a direction opposite to that of the stream or water current, it is said to be moving AGAINST THE STREAM OR CURRENT.

If the boat is moving with a certain speed in water that is not moving, the speed of the boat is then called SPEED OF THE BOAT IN STILL WATER.

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 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 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.

If \(U\) is the speed of the boat down the stream and \(V\) is the speed of the boat up the stream, then we have the following relationship.

Speed of the boat in still water = \[\frac{{U + V}}{2}\]

Speed of the water current = \[\frac{{U - V}}{2}\]

In problems, instead of a boat, it may be a swimmer but the approach is exactly the same. Instead of boats / swimmers in water, it could also be a cyclist cycling against or along the wind. The approach to solving the problems still remains the same.

Example 15: A boat travels 30 km upstream in 5 hours and 100 km downstream in 10 hours. Find the speed of the boat in still water and the speed of the stream.

Solution:

Upstream speed = \(\frac{{{\rm{30}}}}{{\rm{5}}}\)= 6 kmph, Downstream speed is \(\frac{{{\rm{100}}}}{{{\rm{10}}}}\) = 10 kmph.

Speed in still water = \(\frac{{{\rm{6}} + {\rm{10}}}}{{\rm{2}}}\) = 8 kmph

Speed of the stream = \(\frac{{10 - 6}}{2}\) = 2 kmph

Example 16: A cycle track is a right triangle with a difference of 2 km between the legs. Its hypotenuse passes along a side road and the two legs pass along a highway. One of the participants of the cycle race took the side road and raced with the speed of 30 km/h and then he covered the two intervals along the highway during the same time with the speed of 42 km/h. Find the length of the race track.

Solution: Cyclist covered hypotenuse with a speed of 30 km/h and the remaining two sides with the speed of 42 km/h taking same time. Hence ratio of the lengths of hypotenuse and the remaining two sides is 30 : 42 = 10:14. 

Since difference in the sides is 2, and their sum is 14, hence sides are 8 and 6, length of the track  is 24 km.