How Do Cosmic Rays Travel Through Earth's Atmosphere?

AI Thread Summary
Cosmic rays travel through Earth's atmosphere at a speed calculated to be 0.5 times the speed of light, based on a distance of 60 km covered in 400 microseconds. For the cosmic ray, the journey takes approximately 346 microseconds due to time dilation effects. Additionally, the cosmic ray perceives the thickness of the Earth's atmosphere as being about 52 km, influenced by length contraction. The discussion highlights the application of time dilation and length contraction equations in understanding relativistic effects. Understanding these concepts is crucial for solving problems related to cosmic ray travel and relativity.
najisalem2009
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Homework Statement


A cosmic ray travels 60km through the Earth's atmosphere in 400μs, as mesured by experimenters on the ground.
a. Calculate the relative velocity between the cosmic ray's intertial reference grame and the Earth's inertial reference frame.

b. How long does the journey take according to the cosmic ray?

c. According to the cosmic ray, how thick is the Earth's atmosphere?

Homework Equations



Time dilation equation
Δt=Δt'/\sqrt{}1-v²/c2

Length contraction
L=L'\sqrt{}1-v²-c²<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> Answers a) v=0.5c, b) t<sub>ray</sub>=346μs, c) L<sub>ray</sub>=52km<br /> <br /> Im very confused on how to approach this question, can anyone point me in the right direction.
 
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Well, part (a) is just asking you to figure out how fast the cosmic ray is moving as measured by the people on Earth. How do you figure out speed given distance and time?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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