Four athletes P, Q, R, S are participating in a 800 m race. At the start P gives Q, R, S a lead of T seconds. P overtakes Q at the 160 m mark, R at the 200 m mark and S at the 400 m mark. P beats S by 20 sec.R completes the race in 3 minutes. By how many seconds does R beat Q?

Hi Let velocity of P be Vp, Q be Vq, R be Vr and S be Vs and lead given by P to all of them be T seconds Now R takes 180 seconds to complete the race so his speed = 800/180 = 40/9 m/s Now P beats R at 200 m mark So we can say 200/Vp +T =200/Vr Substituting Vr we get 200/Vp +T =45 (1) Now P overtakes S at 400 m mark we get 400/Vp +T =400/Vs (2) Now it is given that P beats S by 20 seconds So as we know at 400m mark both P and S are together so in remaining 400 meters P covers those 400 meters 20 seconds faster than S so we get 400/vp +20 =400/vs (3) From (2) and (3) we get T=20 seconds Substituting in (1) we get Vp= 8m/s Now P beats Q at 160m mark so we get 160/vp +T=160/Vq we get 40=160/Vq so Vq=4m/sec So Time by Q to complete 800 m 800/4=200seconds so R beat Q by 20 seconds