On a statight road a policeman in a jeep chases a thief with a speed v. When the thief is a distance d from the police - jeep , the thief escapes with a bike with a constant acceleration a. The police can catch the thief if ` v^(2) ge 2 ad -` prove it .

Explanation

f the velocity of the bike of the thief increase from 0 to v in time t ,
v= at or , `t= ( v) /(a) `
the distance travelled by the thief in time t ,
` s =(1)/(2) at^(2) =(1)/(2) a ( (v)/( a))^(2) = ( 1)/(2) ( v^(2))/(a)`
the distance travelled by the police - jeep in time t has to be at least ( s+d) .
hence , v `* ((v)/(a)) ge s+d`
or ,` (v^2)/(a ) ge (1)/(2) ( v^(2))/(a) + d or , (1)/(2) (v^(2))/(a) ge d `
`or , v^(2) ge 2ad `