Two horses are approaching one another on a straight tract running east to west.

Question

Two horses are approaching one another on a straight tract running east to west. At t=0 the first horse is plodding along at 1 km/hr. The second horse is moving towards the first at 2 km/hr. When they start out they are 1 km apart. At t=0 a fly started off the nose of the first horse at 3km/hr. When it reached the nose of the second horse it immediately turned around and retouched the first horse, and then turned again to approach the second horse, and so on until the horses met each other.
How far had the first horse traveled when the fly touched the second horse for the first time? How far had the second horse gone at this point? What is the new distance between the horses, at this time? How far had the fly flown on this first leg?
The fly repeats this game, buzzing between the horses until they meet. How far did the fly fly until he was smushed by the meeting noses?

Explanation / Answer

the fly is moving with a speed of 4 km/hr(as it stared off wrt nose which is already moving with 1km/hr so

velocity of fly=3+1=4km/hr)and horse is moving at 2km/hr

and distance between them is 1km

so relatively the fly is moving 6km/hr towards the horse

the time taken by fly to touch second horse is

t=1/6=0.167 hr

so in 0.167 hr the distance travelled by first horse is

d=0.167*1=.167km

so distance travelled by second horse is d=0.167*2=0.334km

distance between the horses now is 1-.167-.334=0.5km

distance travelled by fly is 0.167*4=0.668km

the horses takes 1/3 hr to meet i.e. 0.33hr(since relative velocity=3km/hr and distance to be travelled is 1km so t=d/v=1/3 =0.33hr)

second time 0.5 km distance

so fly relative velocity wrt to first horse is 5km/hr so t=d/5=0.1hr

distance between horses is 0.5-3*0.1=0.2km

so distance travelled by fly is 4*0.1=0.4km

third time 0.2km

so relative velocity of fly wrt to second horse is 6km/hr

so t=0.2/6=0.033hr

distance between horses=0.2-(0.033*3)=0.133km

distance travelled by fly is .033*4=.132km

fourth time 0.133km

so relative velocity of fly wrt first horse is 5km/hr

t=0.133/5=0.0266hr

distance between horses=0.133-(3*0.0266)=0.0532km

distance travelled by fly =4*0.0266=.1064km

fifth distance =0.05032km

so relative velocity of fly wrt to second horse is 6km/hr

t=.05032/6=0.0083hr

distance between horses=0.0532-(3*0.0083)=0.0283km

distance travelled by the fly=4*0.0083=0.0332km

sixth distance is 0.028km

vrelative=5km/hr

t=0.028/5=0.006hr

distance between horses=0.028-(3*0.006)=0.010km

distance travelled by the fly=4*0.006=0.024km

secenth distance is 0.010km

vrel=6km/hr t=0.01/6=0.002

distance between horses=.010-(0.002*3)=0.004km

distance travelled by the fly=4*0.002=0.008km

eight distance is 0.004km

vrel=5km/hr t=.004/5=0.001hr

distance between horses=.004-(.001*3)=0.001km

distance travelled by fly=4*.001=0.004km

sice the distance after this s so small we can neglect it

so total distance travelled byfly=0.004+0.008+0.024+0.0332km+.1064km+.132km+0.4km+0.668km=1.3756km

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