An unstable particle is created in the upper atmosphere from a cosmic ray and tr

Question

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the Earth with a speed of 0.99540c relative to the Earth. A scientist at rest on the Earth’S surface measures that the particle is created at an altitude of 45.0 km.

(a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the Earth?

(b) Use the length contraction formula to calculate the distance from where the particle is created to the surface of the Earth as measured in the particle’s frame.

(c) In the particle’s frame, how much time elapses from when the particle is created to when it strikes the surface of the Earth?

(d) Calculate this time both by the time dilation formula and also from the distance calculated in part (b).

Explanation / Answer

Calculate gamma from the speed. gamma = 1/sqrt(1- (v/c)^2) = 1/sqrt(1 - (.99540)^2) = 10.437731 (carrying extra sig figs here) (a) 45.0 / c (with c as 299792.458) (b) 45.0 / gamma (above) (c) it agrees with its speed, it just measures a shorter distance travelled ... so "how far it went" / "its speed" = "the time it took"

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