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Relative Velocity

To introduce the concept of relative velocity, consider two trains moving on two straight and parallel tracks with same speed and in the same direction. Although both the trains are in motion, with respect to trees, buildings, etc., along the two sides of the tracks, yet to the observer of one train, the other train does not seem to be moving at all. In other words, the velocity of the train appears to be zero.

Let two objects P and P’ be moving with uniform velocities v and v’ along two straight and parallel tracks. Let x0 and x0 be their distances from the origin at t = 0 (initially). If at any time t, x and x’ are the positions (distances) of the two objects respectively with respect to the origin of the position-axis, then for the object, P,
x = x0 + vt                                --------- 3.19
And for the object P’.
x’ = x0 + v’t                              ---------- 3.20
Subtracting equation (3.20) from (3.19), we have
x’ – x = (x0 – x0) + (v’ – v)t           ---------- 3.21
The above equation gives displacement of the object P’ from the object P at any time t. The relative displacement i.e. x’ – x may be positive, zero or negative.
  1. when x’ – x is positive: It means that the object P’ is to the right of the object P.
  2. When x’ – x is zero: It implies that the both objects P’ and P exactly coincide with each other.
  3. When x’ – x is negative: It indicates that the object P’ is to the left of the object P.

Further, in equation (3.21), v’ – v is the relative velocity of the object P’ with respect to the object P. The relative velocity v’ – v may also be positive, zero or negative.
  1. When v’ – v is positive: The equation (3.21) tells that relative distance between the two objects will increase by an amount v’ – v after each unit of time.
  2. When v’ – v is zero. For v’ – v = 0, the equation (3.21) reduces to X’ – x = x0 – x0
    i.e. two objects will remain always at the same constant distance from each other, which is equal the relative distance between them initially (at 't' = 0).
  3. When v’ – v is negative: The equation (3.21) tells that the distance between the two objects will go on decreasing by the amount v – v’ after each unit of time. After some time, the two objects will meet or come together and then the object P’, which was to the right of P will get more and more to the left of P.

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