Question

The velocity components of a particle moving in the $xy$ plane of the reference frame $K$ are equal to ${v_x}$ and ${v_y}$ . Find the velocity $v'$ of this particle in the frame $K'$which moves with the velocity $V$ relative to the frame $K$ in the positive direction of its $x$ axis.

Answer 1

Hint: The velocity components in the $xy$ planes are given with reference to some other frame. We can directly use the velocity addition formula to find the velocity $v'$ along X-axis and Y-axis. Then using the formula for resultant velocity, we can calculate the velocity $v'$ in the frame $K'$ .

Complete step by step answer:
Lets first write down the given quantities:
The velocities in the $xy$ plane with reference frame $K$ is:
${v_x}$ and ${v_y}$ .
We have to find the velocity $v'$ with respect to frame $K'$ which is moving with a velocity $V$ with respect to $K$ frame. Please note the frame $K'$ is moving in the positive $x$ axis with respect to $K$ frame.
Let’s apply vector addition formula, we have
${v_x}' = \dfrac{{{v_x} - V}}{{1 - \dfrac{{V{v_x}}}{{{c^2}}}}}$ and ${v_y}' = \dfrac{{{v_y}\sqrt {1 - \dfrac{{{V^2}}}{{{c^2}}}} }}{{1 - \dfrac{{V{v_x}}}{{{c^2}}}}}$
Where $c$ is the speed of light;
Now the resultant velocity will be given as:
$v' = \sqrt {{v_x}{'^2} + {v_y}{'^2}} $
Substituting the values from above, we get
$v' = \dfrac{{\sqrt {{{\left( {{v_x} - V} \right)}^2} + {v_y}^2\left( {1 - \dfrac{{{V^2}}}{{{c^2}}}} \right)} }}{{1 - \dfrac{{{V^2}}}{{{c^2}}}}}$
This is the required velocity of the particle.

Additional Information:
We get different velocities because the two observers are in different frames of reference. A frame of reference is a set of coordinates that can be used to determine positions and velocities of objects in that frame, these coordinates are different when observed from different frames of reference. Two frames moving with constant velocity will have the same observations. In physics, there is no absolute frame of reference with respect to ‘Laws of Physics’. Special relativity arises when any two inertial frames are equivalent. A frame moving with constant velocity with respect to an inertial frame is also an inertial frame.

Note:
If the frame moves with speed of light then the particle’s speed will be infinitely very high. Two frames having the same velocity or constant velocity will have the same observations. When different frames do not have constant velocity then different observations are observed.