With what velocity (relative to the reference frame $K$) did the clock move, if during the time interval $t=5.0s$, measured by the clock of the frame $K$, it became slow by $\Delta t=0.10s$?

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. 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. A rod flies with constant velocity past a mark which is stationary in the reference frame $K$. In the frame $K$ it takes $\Delta t=20ns$ for the rod to fly past the mark. In the reference frame fixed to the rod the mark moves past the rod for $\Delta t^{\prime }=25ns$. Find the proper length of the rod.

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?