Calculating Light Signal Velocity: Relativistic Effects at 0.5c

AI Thread Summary
To calculate the velocity components of a light signal fired at 60° north of west, the observer moving eastward at 0.5c can use the relativistic velocity addition formulas. The first formula, U' = (u-v)/(1-uv/c), helps find the components of the light signal's velocity. The angle complicates the calculation, requiring the use of trigonometric functions to resolve the signal's velocity into x and y components. A similar approach applies for the observer moving westward, utilizing the second formula, U = (u+v)/(1+uv/c). Properly applying these formulas and trigonometry will yield the magnitude and direction of the light signal's velocity for both observers.
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Homework Statement



You fire a light signal at 60° north of west. (a) find the velocity components of this signal according to an observer moving eastward relative to you at 0.5c. From them, determine the magnitude and direction of light signal's velocity according to the other observer. (b) find the components according to a different observer, moving westward relative to you at 0.5c.

Homework Equations



U'= (u-v)/(1-uv/c)
U= (u+v)/(1+uv/c)

The Attempt at a Solution



These are the only two formulas that I could think to use, but I am not sure how to incorporate the angle? It's throwing me off of the whole problem. Any help or explanation would be so much appreciated, thank youu!
 
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Would I use the speed of light for the hypotenuse of my triangle, since it says I'm firing a light signal, and then use simple trig, i.e. Csin60=y for my y-component? I'm really unsure.
 
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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