# 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 Pâ€™ be moving with uniform velocities v and vâ€™ along two straight and parallel tracks. Let x

_{0}and x

_{0}

^{â€™}be their distances from the origin at t = 0 (initially). If at any time t, x and xâ€™ are the positions (distances) of the two objects respectively with respect to the origin of the position-axis, then for the object, P,

x = x

_{0}+ vt --------- 3.19

And for the object Pâ€™.

xâ€™ = x

_{0}

^{â€™}+ vâ€™t ---------- 3.20

Subtracting equation (3.20) from (3.19), we have

xâ€™ â€“ x = (x

_{0}â€“ x

_{0}) + (vâ€™ â€“ v)t ---------- 3.21

The above equation gives displacement of the object Pâ€™ from the object P at any time t. The relative displacement i.e. xâ€™ â€“ x may be positive, zero or negative.

- when xâ€™ â€“ x is positive: It means that the object Pâ€™ is to the right of the object P.
- When xâ€™ â€“ x is zero: It implies that the both objects Pâ€™ and P exactly coincide with each other.
- When xâ€™ â€“ x is negative: It indicates that the object Pâ€™ is to the left of the object P.

- When vâ€™ â€“ v is positive: The equation (3.21) tells that relative distance between the two objects will increase by an amount vâ€™ â€“ v after each unit of time.
- When vâ€™ â€“ v is zero. For vâ€™ â€“ v = 0, the equation (3.21) reduces to Xâ€™ â€“ x = x
_{0}^{â€™}â€“ x_{0 }

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). - When vâ€™ â€“ v is negative: The equation (3.21) tells that the distance between the two objects will go on decreasing by the amount v â€“ vâ€™ after each unit of time. After some time, the two objects will meet or come together and then the object Pâ€™, which was to the right of P will get more and more to the left of P.