Motion of masses connected by a string over a pulley

[neilparker62]

ii) Impulse on ground when mass B impacts: 2mΔv ?

 

[songoku]

ii) Impulse on ground when mass B impacts: 2mΔv ?

Yes, sorry. The mass should be 2m.For (v) :
Distance of B = twice the distance when it accelerates upwards + decelerate to stop (when A hits the ground)

Speed of B when A hits the ground
$$v^2 = (\frac{1}{3} \sqrt{\frac{2gh}{3}})^2 + 2 . \frac{g}{3} . 2h$$
$$v^2 = \frac{38gh}{27}$$

Distance traveled by B when decelerating:
$$v^2 = u^2 - 2gd$$
$$0 = \frac{38gh}{27} - 2gd$$
$$d = \frac{19h}{27}$$

So total distance
= $$4h + 2 . \frac{19h}{27}$$
= $$\frac{146h}{27}$$

For (vi), I think I need to use formula of sum to infinity of geometric progression but I can not find the ratio. The distance traveled by B:
a. when released from rest until hits the ground = h
b. between first hit and second hit = $$\frac{146h}{27}$$
c. between second hit and third hit (repeating the same process as (v)) = $$\frac{1298h}{243}$$

How to find the total distance?

Thanks