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. 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. In a triangle the proper length of each side equals $a$. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity $V$ along one of its sides.

Q. In a triangle the proper length of each side equals $a$. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity $V$ along one of its bisectors :

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. Find the proper length of a rod if in the laboratory frame of reference its velocity is $v=\frac{c}{2}$, the length $l=1.00m$, and the angle between the rod and its direction of motion is $\theta ={45}^{\circ }$.