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 $v^{\prime }$ of this particle in the frame $K^{\prime }$ which moves with the velocity $V$ relative to the frame $K$ in the positive direction of its x axis.

Q. The reference frame $K^{\prime }$ moves in the positive direction of the x axis of the frame $K$ with a relative velocity $V$. Suppose that at the moment when the origins of coordinates $O$ and $O^{\prime }$ coincide, the clock readings at these points are equal to zero in both frames. Find the displacement velocity $\dot{x}$ of the point (in the frame $K$) at which the readings of the clocks of both reference frames will be permanently identical. Demonstrate that $x<V$.

Q. A rod moves lengthwise with a constant velocity $v$ relative to the inertial reference frame $K$. At what value of $v$ will the length of the rod in this frame be $\eta =0.5\%$ less than its proper length?

Q. In the reference frame $K$ two particles travel along the $x$ axis, one of mass $m_1$ with velocity $\overrightarrow{v_1}$, and the other of mass $m_2$ with velocity $\overrightarrow{v_2}$.Find the velocity $\overrightarrow{v}$ of the reference frame $K^{\prime }$ in which the cumulative kinetic energy of these particles is minimum.

Q. The reference frame, in which the centre of inertia of a given system of particles is at rest, translates with a velocity $\overrightarrow{V}$ relative to an inertial reference frame $K$. The mass of the system of particles equals $m$, and the total energy of the system in the frame of the centre of inertia is equal to $\tilde{E}$. The total energy $E$ of this system of particles in the reference frame $K$ is given as $E=\tilde{E}+\frac{1}{x}m{V}^2$. Find $x$

Q.

A plane sound wave is travelling in a medium. In reference to a frame A, its equation is $y=acos(\omega t-kx)$. Which reference to frame B, moving with a constant velocity v in the direction of propagation of the wave, equation of the wave will be