Tuesday, October 14, 2014

Relative Velocity and Motion of a Boat across a River

Relative velocity is the comparative velocity of one body with respect to other body. One body will have a relative velocity with respect to other one only when there is a relative motion between them. If two bodies are having the motion in the same direction relative velocity of one with respect to other is the difference between them.It is simply because one bodies having more velocity when compared with the other body and how much more can be obtained only by subtracting that from the other value.

If two bodies are travelling in the opposite direction we can incorporate the same concept but being the bodies are in the opposite direction the resultant will become automatically the some of the two vectors as shown.

If two vectors are having an angle between them to find the relative velocity between them we shall incorporate a third body in between them. Generally the third body is the ground.

Expressing velocity of the any body with respect to ground is more appropriate than expressing the velocity of the ground with respect to other body.It is simply because it is not the ground that is moving but the body moving on the ground.

Problem and Solution

Basing on a relative motion of two bodies. Let us consider two bodies each having the same velocity 10 m/s,one moving along the east other moving along the north from the same point. Find the relative velocity of one body with respect to other body?

While we are solving the problem we need to take a reference and the ground into consideration as reference.After identifying the answer,we can identify the direction of the vector.


The person is walking in the rain feel that the velocity of the rain is as twice as his velocity. It which angle you should hold his umbrella with vertical if he’s moving in a forward direction and training is happening in a vertical downward direction therefore he cannot be drenched in the rain?


This problem can also be solved basing on the concept of relative velocity as shown below. Whenever requirement of the third body is there we always get the ground into consideration because it is always there. We prefer to say the velocity of the body with respect to ground than in the  reverse way because bodies move on the ground.

Motion of a boat in a river

There are four different possible pace of about moving across the river.

Case one

When a boat is crossing the river along the direction of the river:

In this case the motion of the boat is bit easy because it is supported by the stream of water therefore the boat takes less time to cross the river.

Case two

Let us consider a case that the boat is moving against the steam of the river. In this case as boat has to overcome the opposition of the river it takes more time.

Case three

Let us consider a case boat has to cross the river in such a way that it has to reach the exact opposite position.In this case we are actually not going along the river but where crossing the river.

If you travel straight to the opposite point as the river pushes you are not going to reach the exact opposite point. That’s why we shall drive river boat with an angle  with the vertical.

In the following derivation we have discussed the that with what angle he shall drive therefore he will be reaching the exact opposite point.In this case you are going to reach the exact opposite point means the path is the shortest but it is going to take the longest time to cross the river.

Case four

In this case boat will go straight to the opposite point and being the river is going to push it it is not going to reach the exact opposite point but some other point in the bank of the river. In this case the party is going to be longest but the time is going to be shortest.

So here we have two choices.When you want to cross the river and the shortest path we have two choose case three where as when you want to cross the river with the shortest time we had to choose case four.

Case five

Suppose you are crossing the river in such a way that you are making an angle  with the vertical, but it is not sufficient to reach the exact opposite position.

In this case we have resolve the component of velocity of the boat along a horizontal and vertical parts. The vertical part will help you to identify the time taken to cross the river whereas the horizontal component of velocity has to be subtracted from the velocity of the river value while calculating the drift of the water. Drift is simply a particular value of the displacement because of which the boat has missed the exact opposite position. The equation for this drift is as shown.

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