Irodov Problem 1.349

Let the original length of the rod (in a frame fixed to the rod) be l0. Also, let the length of the scale's unit be 1 as measured in the frame fixed to the rod.
Suppose that the rod is moving with velocity v relative to the scale.

In the reference frame fixed to the scale:

As seen from the scale, since the rod is moving it would appear smaller. Its length would be . Thus, as measured by the scale, the length of rod will be



In the reference frame fixed to the rod:

As seen from the frame fixed to the rod, the rod is stationary and has a length of l0 , however, the scale is moving with a velocity v. Thus, each unit of the scale will shrink and will be . Thus, when measured using the scale the measured length will be,




From (1) and (2) we have,

Irodov Problem 1.348

Since the two particles a traveling at the same speed v, relative to each other they are stationary. This means that as measured from a reference frame fixed to any of the particles, the distance between them l0 will be the original distance. The distance measured by someone in the Laboratory will however be a shrunk version of the original distance between the particles given by . Similarly, time will appear to be slower in the reference frame fixed to the particles compared to the time measured in the Laboratory. So while the time interval between collisions measured in the Laboratory reference frame is , the same however, when measured in the reference frame fixed to the particles will be a lesser (since time is slower) given by .

Solution to the Problem in Reference Frame of the Particles:

In the reference frame of the particles we have,








Solution to the Problem in the Laboratory Reference Frame :

Irodov Problem 1.347

This problem is very similar to Problem 1.346. As seen from the Laboratory reference frame, the Muon's time would be dilated (lowed down) and appear longer. Let the proper decay time of the Muon be . In its own reference frame the Muon will decay in time , however as seen in the Laboratory it will decay in a longer time given by. The distance l traveled by the Muon in the Laboratory reference frame is given by,








b) The distances in the Muon's reference frame will seem to be shrunk as compared to Laboratory and so it will feel it has traveled a much shorter distance given by .

Irodov Problem 1.346

The Particle moves very fast compared to someone in the Laboratory. Consequently, time will appear to be slower for the fast moving particle as seen from Laboratory reference frame. While in its own reference frame, the particle will decay exactly within its correct lifetime , to someone in the Laboratory reference frame this will appear to be a longer interval . The relation is given by,


This means that as seen in Laboratory, the particle would have traveled a distance of,