On a cricket field, the batsman is at the origin of coordinates and a fielder stands in position given as ( 46 i ^ + 28 j ^ ) m . The batsman hits the ball so that it rolls along the ground with con

Question

On a cricket field, the batsman is at the origin of coordinates and a fielder stands in position given as ( 46 i ^ + 28 j ^ ​ ) m . The batsman hits the ball so that it rolls along the ground with constant velocity given by ( 7.5 i ^ + 10 j ^ ​ ) m / s . The fielder can run with a speed of 5 m / s . If he starts to run immediately the ball is hit, what is the shortest time in which he could intercept the ball.

TestHub · Accepted Answer

The ball is at ( 7.5 t , 10 t ) ; interception needs ( 7.5 t − 46 ) 2 + ( 10 t − 28 ) 2 = 25 t 2 , i.e. 131.25 t 2 − 1250 t + 2900 = 0 or 21 t 2 − 200 t + 464 = 0 . Roots: t = 42 200 ± 40000 − 38976 ​ ​ = 42 200 ± 32 ​ , giving t = 4 s (shortest) or 5.52 s. Matches the printed answer.

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