An object is dropped from height h from the ground. Every time it hits the ground it loses 50% of its energy.

Since the object loses half its energy each time it hits the ground. Hence it will rise half of the previous height each time, as potential energy at height h is given by mgh (proportional to h).Hence it will rise h/2 after the first hit, then h/4 and so on.Hence total distance traveled will beS=h+(h/2+h/2+h/4+h/4+..)=h+(h+h/2+)=h+h(1+1/2+1/4+..)=h+h(1+11𕒵2)= 3h