Question

Define relative velocity. Explain in detail the concept of relative velocity.

Answer 1

Relative velocity. Relative velocity of an object A with respect to another object B, is the time rate at which the objects A changes its position with respect to the object B.

To understand the concept of the relative velocity, consider two objects A and B, moving uniformly with velocities v₁ and v₂ (fig).

Let x₀₁ = displacement of A at t = 0

x₀₂ = displacement of B at t = 0

 Now, let x₁ and x₂ be the displacement of A and B respectively after time t respectively.

So, the displacement of A in time t is given by:

(x₁ - x₀₁) =v₁t or x₁ = x₀₁ + v₁t.......(i)

Similarly, the displacement of B in time t is given by

(x₂ - x₀₁) = v₂t or x₂ = x₀₂ + v₂t.....(ii)

Subtracting eqn.(i) from eqn. (ii) we get,

(x₂ - x₁) = (x₂₀ +x₁₀) + (v₂ -v₁)t.......(iii)

Now, (x₂₀ - x₁₀) = x₀, the relative displacement of B w.r.t. A at t = 0

(x₂ - x₁) = x, the relative displacement of B w.r.t A at instant t.

Hence, equation (iii) can be written as

x = x₀ + (v₂ -v₁)t

or, (v₂ -v₁) = (x -x₀)/t......(iv)

Which is the relative velocity of object B,with respect to the object A.

Thus, relative velocity of the object B w.r.t., the object A,

v_BA = v₂ -v₂.

Special Cases:

(1) When the two objects move with equal velocities:

i.e., v₁ = v₂ or v₂ - v₁ = 0, then from eqn. v = (iv)

x - x₀ = 0, or x = x₀.

It means the two object stay at constant distance apart during the whole journey. In this case, the position-time graphs of two objects are parallel straight lines as shown in the figure 3.38 (a).

(2) When the two objects move with unequal velocities:

(i) when v₁ > v₂ then equation (iv), (x -x₀) is negative. This shows that the separation between two moving objects will go on decreasing with time. After Sometime, the two moving object will  meet and then the object moving with velocity v₁ (i.e., faster) will move more and more to the left of slow moving object as shown in the figure (b)

(ii)  When _v2 > v₁, then eqn, (iv) (x -x₀) is positive. This shows that the separation between two moving objects will go on increasing with time as shown in fig (c).

(3) When tow objects move in opposite directions:

Let object A moves along positive X direction with velocity v₁ and object B moves along negative X direction with velocity v_2.

Hence, relative velocity of object A w.r.t., the object B is given by v_AB = v₁-(-v₂) = v₁ +v₂.

 If both the objects A and B move along parallel straight lines in the same direction, then relative velocity of A w.r.t B is given by 

v_AB = v₁ - v₂

and the relative velocity of B w.r.t., A is given by

v_BA = v₂ -v₁.

If both the objects A and B move along parallel straight lines in the opposite directions then relative of A w.r.t., B is given by

v_AB = v₁ - (v₂) = v₁ +v₂

and the relative velocity of B w.r.t., A is given by 

v_BA = v₂ +v₁.