In the laboratory we are observing the motion of a particle moving in the positive direction of the axis $x$ of a frame system attached to it. At the initial moment, the particle is in the origin and in a time interval of $2.0$ ns covers a distance of $25$ cm. A spacecraft passes with velocity $v = 0.80c$ along the $x$ direction of the laboratory, in the positive direction, and from it observes the motion of the same particle. Determine the average particle speeds in the two systems of reference. What interval of time and what distance would an observer placed on the spaceship?
Why can't I use the formulas of time expansion and length contraction, but instead I must use the Lorentz transformations?