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Relative velocity

What is relative velocity?

Easy but tricky question!

Easy, because it is the difference of the velocities of two bodies.

Tricky, because velocity is a vector quantity and the difference of velocities must be calculated as vectors.

Take some examples.

(1) Suppose you are driving a car, on a straight road with a velocity of 90 km/hr. Let your friend is leading you in another car and s/he is also going at 90 km/hr. Is the distance between your cars increasing or decreasing? No, because your relative velocity is zero (although you are moving with 90 km/hr.). How did we calculate this? Simply, 90 km/hr – 90 km/hr = 0 km/hr. Let’s do it in terms of vectors.

This is your velocity vector.

This is the velocity vector of your friend. In the same direction and magnitude.

Now to find the difference, we subtract CD from AB. We know we have to take negative vector of CD; say vector DC with the same magnitude but opposite direction.

And now add this to the velocity vector of your car, AB by head-to-tail rule. We see the resultant is zero.

(2) Let on some other day, you are again driving on the same road at 90 km/hr. One another car driver overshoot you. He is probably going on 120 km/hr. What is his velocity with respect to you (or relative to you)? Obviously, 120 – 90 = 30 km/hr. Again subtraction. (Do it as vectors yourself). What is your velocity in relation to him? 90 – 120 = – 30 km/hr. Thus he feels you are going BACK.

(3) Now take the interesting case when both cars are approaching one another. Let you are going at 120 km/hr and the other car is coming to you from the opposite side at 90 km/hr. What is the relative velocity of the approaching car, or saying more exactly, at what velocity does the other car approaches you? 120 + 90 = 210 km/hr. Is this a sum this time? NO. This is also subtraction. In fact, due to its opposite direction, the velocity of the other car is – 90 km/hr. When we subtract it, then 120 – (- 90) = 120 + 90 = 210 km/hr.

In vector form your velocity of 120 km/hr. is represented by the following vector.

Similarly, the velocity of the other car is 90 km/hr in opposite direction.

Now the relative velocity is the difference of the velocities of the two bodies. Therefore, subtract the second velocity from the first one. Therefore, take the negative vector of velocity CD.

Now add the two vectors to find their difference.

(This is relative velocity of approaching and is given by v1- v2.)

(4) Take another example and this time you make a try. Consider two cars, coming from opposite sides. Let they cross one another and now they go in opposite directions back to back. How will you find their relative velocity?

Hint: (1) Addition of the magnitudes of the velocities.

(2) This is called relative velocity of separation. 

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