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