Answer:
Part a)
Velocity of sled
![v = \frac{m_1 s}{m_1 + m_2 + M}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D)
velocity of first man who jump off
![v_1 = -\frac{(m_2 + M) s}{m_1 + m_2 + M}](https://tex.z-dn.net/?f=v_1%20%3D%20-%5Cfrac%7B%28m_2%20%2B%20M%29%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D)
Part b)
Velocity of sled
![v_f = (\frac{m_1 s}{m_1 + m_2 + M}) + (\frac{m_2}{m_2 + M})s](https://tex.z-dn.net/?f=v_f%20%3D%20%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20%2B%20%28%5Cfrac%7Bm_2%7D%7Bm_2%20%2B%20M%7D%29s)
Also the speed of second person is given as
![v_2 = (\frac{m_1 s}{m_1 + m_2 + M}) - \frac{Ms}{m_2 + M}](https://tex.z-dn.net/?f=v_2%20%3D%20%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20-%20%5Cfrac%7BMs%7D%7Bm_2%20%2B%20M%7D)
Part c)
change in kinetic energy of sled + two people is given as
![KE = \frac{1}{2}Mv_f^2 + \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B1%7D%7B2%7DMv_f%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dm_1v_1%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dm_2v_2%5E2)
Explanation:
As we know that here we we consider both people + sled as a system then there is no external force on it
So here we can use momentum conservation
since both people + sled is at rest initially so initial total momentum is zero
now when first people will jump with relative velocity "s" then let say the sled + other people will move off with speed v
so by momentum conservation we have
![0 = m_1(v - s) + (m_2 + M)v](https://tex.z-dn.net/?f=0%20%3D%20m_1%28v%20-%20s%29%20%2B%20%28m_2%20%2B%20M%29v)
![v = \frac{m_1 s}{m_1 + m_2 + M}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D)
so velocity of the sled + other person is
![v = \frac{m_1 s}{m_1 + m_2 + M}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D)
velocity of first man who jump off
![v_1 = \frac{m_1 s}{m_1 + m_2 + M} - s](https://tex.z-dn.net/?f=v_1%20%3D%20%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%20-%20s)
![v_1 = -\frac{(m_2 + M) s}{m_1 + m_2 + M}](https://tex.z-dn.net/?f=v_1%20%3D%20-%5Cfrac%7B%28m_2%20%2B%20M%29%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D)
Part b)
now when other man also jump off with same relative velocity
so let say the sled is now moving with speed vf
so by momentum conservation we have
![(m_2 + M)(\frac{m_1 s}{m_1 + m_2 + M}) = m_2(v_f - s) + Mv_f](https://tex.z-dn.net/?f=%28m_2%20%2B%20M%29%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20%3D%20m_2%28v_f%20-%20s%29%20%2B%20Mv_f)
![(m_2 + M)(\frac{m_1 s}{m_1 + m_2 + M}) + m_2s = (m_2 + M)v_f](https://tex.z-dn.net/?f=%28m_2%20%2B%20M%29%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20%2B%20m_2s%20%3D%20%28m_2%20%2B%20M%29v_f)
Now we have
![v_f = (\frac{m_1 s}{m_1 + m_2 + M}) + (\frac{m_2}{m_2 + M})s](https://tex.z-dn.net/?f=v_f%20%3D%20%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20%2B%20%28%5Cfrac%7Bm_2%7D%7Bm_2%20%2B%20M%7D%29s)
Also the speed of second person is given as
![v_2 = (\frac{m_1 s}{m_1 + m_2 + M}) + (\frac{m_2}{m_2 + M})s - s](https://tex.z-dn.net/?f=v_2%20%3D%20%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20%2B%20%28%5Cfrac%7Bm_2%7D%7Bm_2%20%2B%20M%7D%29s%20-%20s)
![v_2 = (\frac{m_1 s}{m_1 + m_2 + M}) - \frac{Ms}{m_2 + M}](https://tex.z-dn.net/?f=v_2%20%3D%20%28%5Cfrac%7Bm_1%20s%7D%7Bm_1%20%2B%20m_2%20%2B%20M%7D%29%20-%20%5Cfrac%7BMs%7D%7Bm_2%20%2B%20M%7D)
Part c)
change in kinetic energy of sled + two people is given as
![KE = \frac{1}{2}Mv_f^2 + \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B1%7D%7B2%7DMv_f%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dm_1v_1%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dm_2v_2%5E2)
here we know all values of speed as we found it in part a) and part b)