Question

how many times will the half life of an unstable particle increases as observed by an observer on the Earth if the particle moves with a velocity of 0.99C with respect to the observer
1)4 times
2)5 times
3)6 times
4)7 times

Answer 1

Lorentz transformations predict the following relation between the time intervals in the moving (with the object) and laboratory (that is "at rest") frames (i.e., the time dilation phenomenon):

$$\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}},$$

where \Delta t₀ is the time interval in the frame of the moving unstable particle and \Delta t is the time interval in the laboratory frame. Hence, the increase in the half-life period of the unstable particle in respect to the laboratory frame can be obtained as follows:

$$\frac{\Delta t}{\Delta t_0} = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{1}{\sqrt{1-0.99^2}} \approx 7$$

Answer: 4) 7 times.