How Does Small y and d Affect the Difference Between r2 and r1 in Physics?

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The discussion revolves around the mathematical derivation of the difference between r2 and r1 in a physics context, specifically under the conditions where both y and d are much smaller than L. The initial equations for r1 and r2 are provided, leading to the expression r2 - r1. The user initially struggles with the problem but later finds a solution using the binomial approximation. Another participant encourages sharing the solution for the benefit of others facing similar challenges. Sharing insights can enhance collective understanding in physics problem-solving.
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Homework Statement



r1 = √(L2+(y-0.5d)2)
r2 = √(L2+(y+0.5d)2)
Show that in the limit that both y and d are <<L r2 - r1 = yd/L

Homework Equations


The Attempt at a Solution



r2 - r1 =(√(L2+(y+0.5d)2)) - (√(L2+(y-0.5d)2))

Nevermind, I've figured it out but can't seem to be able to delete this post :s
 
Last edited:
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I'd have tried substituting x=[(y-0.5d)/L]2 and used the binomial approximation.
Oh you figured it out - rather than delete the post - how about posting your result: that way others who get stuck can benefit.
 
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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