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đ r C A đ t = đ r B A đ t + đ r C B đ t

v C A = v B A + v C B

The meaning of the subscripted velocities are :

  • v C A : velocity of object "C" with respect to "A"
  • v C B : velocity of object "C" with respect to "B"
  • v B A : velocity of object "B" with respect to "A"

Problem : A boy is riding a cycle with a speed of 5√3 m/s towards east along a straight line. It starts raining with a speed of 15 m/s in the vertical direction. What is the direction of rainfall as seen by the boy.

Solution : Let us denote Earth, boy and rain with symbols A, B and C respectively. The question here provides the velocity of B and C with respect to A (Earth).

v B A = 5 3 m / s v C A = 15 m / s

We need to determine the direction of rain (C) with respect to boy (B) i.e. v C B .

v C A = v B A + v C B

v C B = v C A - v B A

We now draw the vector diagram to evaluate the terms on the right side of the equation. Here, we need to evaluate “ v C A - v B A ”, which is equivalent to “ v C A + ( - v B A ) ”. We apply parallelogram theorem to obtain vctor sum as reprsesented in the figure.

Relative velocity

For the boy (B), the rain appears to fall, making an angle “θ” with the vertical (-y direction).

tan θ = v B A v C A = 5 3 15 = 1 3 = tan 30 0

θ = 30 0

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Acceleration of the point object

If the object being observed is accelerated, then its acceleration is obtained by the time derivative of velocity. Differentiating equation of relative velocity, we have :

đ v C A đ t = đ v B A đ t + đ v C B đ t

a C A = a B A + a C B

The meaning of the subscripted accelerations are :

  • a C A : acceleration of object "C" with respect to "A"
  • a C B : acceleration of object "C" with respect to "B"
  • a B A : acceleration of object "B" with respect to "A"

But we have restricted ourselves to reference systems which are moving at constant velocity. This means that relative velocity of "B" with respect to "A" is a constant. In other words, the acceleration of "B" with respect to "A" is zero i.e. a B A = 0 . Hence,

a C A = a C B

The observers moving at constant velocities, therefore, measure same acceleration of the object (C).

Interpretation of the equation of relative velocity

The interpretation of the equation of relative motion in two dimensional motion is slightly tricky. The trick is entirely about the ability to analyze vector quantities as against scalar quantities. There are few alternatives at our disposal about the way we handle the vector equation.

In broad terms, we can either use graphical techniques or vector algebraic techniques. In graphical method, we can analyze the equation with graphical representation along with analytical tools like Pythagoras or Parallelogram theorem of vector addition as the case may be. Alternatively, we can use vector algebra based on components of vectors.

Our approach shall largely be determined by the nature of inputs available for interpreting the equation.

Equation with reference to earth

The equation of relative velocities refers velocities in relation to different reference system.

v C A = v B A + v C B

We note that two of the velocities are referred to A. In case, “A” denotes Earth’s reference, then we can conveniently drop the reference. A velocity without reference to any frame shall then mean Earth’s frame of reference.

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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