Question

A thief runs away from a police station with a uniform speed of 100m/minutes. After one minute a policeman runs behind the thief to catch him at speed of 100m/minute in first minute & increases his speed 10m each succeeding minute. After how many minutes, the policeman will catch the thief?

Answer 1

Let n be the number of minutes after which policeman catches the thief.

then at that instant:

Distance travelled by thief = Distance travelled by policeman

Distance travelled by thief:

Since the thief moves at a constant speed of 100m per min, distance travelled by him = 100(n+1)

Distance travelled by policeman:

Now the policeman increases his speed by 10m per minute after starting with an initial speed of 100m per minute.

Distance travelled by policeman in first minute: 100

Distance travelled by policeman in second minute: 110

Distance travelled by policeman in third minute: 120

Distance travelled by policeman in fourth minute: 130

As we can clearly see this is an A.P with first term 100 and common difference 10.

Hence, Total distance travelled by police in n minutes = 100 +110 + 120 +......= n/2(2*a+(n-1)d)

Now, Total distance travelled by thief in n minutes = Total distance travelled by policeman in n minutes.

100 (n+1) = n/2 (2*100 + (n-1)10)
100*n + 100 = 100*n + 5n2 - 5*n
5n2 - 5*n - 100 = 0
n2 - n - 20 = 0
n2 + 4*n - 5*n - 20 = 0
n ( n + 4 ) - 5 ( n + 4 ) = 0
( n - 5 )( n + 4 ) = 0
Hence, n = 5 or n = -4.

But n > 0, ( n is the number of minutes )

Therefore n = 5.

Hence, the policeman catches the thief 5 minutes after the policeman starts running.

Answer 2

600m
Seee the solution time =5min