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A shell moving through the air is split into two fragments of mass m1 and m2 by its explosion which adds on energy E to the fragments. The magnitude of relative velocity between the two fragments after explosion is:

  1. $\sqrt{\frac{2E(m_1+m_2)}{(m_1m_2)}}$
  2. $\sqrt{2E}\frac{(m_1+m_2)}{(m_1m_2)}$
  3. $\sqrt{2E}\left[\frac{\sqrt{m_1}+ \sqrt{m_2}}{\sqrt{m_1m_2}}\right]$
  4. $0$

My progress so far:

  • Shell exploded, fragments gained energy, so they can't stay stuck together. Therefore, option 4 i.e. $v_{rel}=0$ , is wrong.
  • Option 2 is dimensionally wrong.
  • Option 1 & option 3 differ in only numerator, by: $\sqrt{m_1+m_2} \;\; vs.\;\; (\sqrt{m_1} + \sqrt{m_2})$
  • Tried $F_{ext} = 0, \ \ \Delta E = 0$, but couldn't isolate the variables for further solving as it is.
  • I tried clever observation/guessing. If I take $m_1=m_2=m$ , then the relevant options (1, 3) got reduced to $\sqrt{4E/m} \;\; vs. \;\; 2\sqrt{2E/m}$.
  • I will add further progress after making this assumption after i read the notifications for this question.

My Solution after taking the assumption above mentioned (I am not asking for review, just am telling my progress, or work done by me so u dont have to repeat this) :

$$F_{ext}=0 \Rightarrow p_i = p_f\;\;\; \& \;\;\; v_{cm, i} = v_{cm,f}$$ $$\Rightarrow 2mu = m(v_1+v_2) \;\;\; (\because m_1=m_2)$$ $$or, \;\;\; 4u^2 = (v_1+v_2)^2 \;\;\; (\text{dividing by $m$, then squaring}) \;\;\; \ldots(i)$$


$$\Delta E = 0 \Rightarrow K_i + E = K_f$$ $$(\times 4) \Rightarrow 4mu^2 + 4E = 2m(v_1^2+v_2^2)$$ $$or, \;\;\; 4E = m (v_1-v_2)^2 = m(v_{rel}^2) \;\;\; (\text{from $i$, and rearranging})$$ $$or, \;\;\; |v_{rel}| = \sqrt{4E/m}$$ $$\text{which is Option 1 (confirmed answer).}$$


So, I think the problem remains in isolating the variables....

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    $\begingroup$ Have you tried using the energy an momentum conservation laws? $\endgroup$ – Adam Latosiński Feb 14 at 11:34
  • $\begingroup$ yup, but no go... am retrying with some clever modifications to see if they work $\endgroup$ – user8395964 Feb 14 at 11:52
  • $\begingroup$ Please take a minute to read our guidelines for homework and exercise questions as well as check-my-work questions. We intend our questions to be potentially useful to a broader set of users than just the one asking, and we prefer conceptual questions over those just asking for a specific computation. $\endgroup$ – Emilio Pisanty Feb 14 at 12:17
  • $\begingroup$ @emilio-pisanty should i add my pre work in post or as comment? and at the moment i dont have a good quality camera, so will writing the work be good? (i will still try to capture and post my work, but just clarifying in advance)! $\endgroup$ – user8395964 Feb 14 at 12:32
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    $\begingroup$ writing in TeX is really time consuming, more than 1 hour just flew by since I started writing :surprised: $\endgroup$ – user8395964 Feb 14 at 16:50