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physics relative velocity help needed please..

Wind is blowing west to east along 2 parallel tracks .Two trains moving with the same speed in opposite direction have the steam track of one double than the other .the speed of each train is ?

 

  1. 3 times that of the wind
  2. 1/2  times that of the wind
  3. 2 times that of the wind
  4. equal to  that of the wind


     
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1 Answer

First, we identify the motion as 1-dimensional. We can analyze it using a single axis (x - axis). Let's take the positive direction to be towards the right (towards the east).

The velocity of each train's steam track is equal to velocity of the train + the velocity of the wind.
The train moving west to east (Train 1) has a velocity v, where v >= 0.
The train moving east to west (Train 2) has a velocity -v.
The velocity of the wind is w, where w >= 0. It is positive, since it blows from west to east.

The velocity of steam track 1 is equal to the sum of the velocity of Train 1 and the wind velocity, or
V1 = v + w
V2 = -v + w.
Speed is the absolute value of velocity, so
S1 = |v + w| and S2 = |-v + w|. From properties of absolute values, S2 >= S1.

From the problem statement, and what we've just seen about absolute values, we know that we must choose
S1 = 2*S2. Also, since v and w are positive, |v+w| = v+w. This gives us

v + w = 2|-v + w|. There are two possibilities: Either (-v + w) is positive, in which case |-v + w| = -v + w, and
v + w = 2(-v + w). solving for v,
v + w = -2v + 2w
3v = w, or v = w/3 (1/3 times that of the wind). This is not a choice.

Or (-v+w) is negative, in which case |-v + w| = v - w, and
v + w = 2(v - w). Solving for v,
v + w = 2v - 2w.
v = 3w (3 times that of the wind). This is answer 1 from your list.