Two trains leave a railway station at same time. The first train travels due west and the second train due north. The first train travels 5 km/hr faster than the second train. If after 2 hours, they are 50 km apart, find the speed of each train.

Let speed of second train be x km/hr and first be (x + 5) km/hr.

According to figure,

Let O be the position where trains leave

Distance travelled by first train OA = 2(x + 5) km

Distance travelled by second train OB = 2x km

By pythagoras theorem we get,

⇒ AB² = OA² + OB²

⇒ 50² = [2(x + 5)]² + (2x)²

⇒ 50² = 4(x + 5)² + 4x²

⇒ 2500 = 4(x² + 25 + 10x) + 4x²

⇒ 2500 = 4x² + 100 + 40x + 4x²

⇒ 8x² + 40x - 2400 = 0

⇒ 8(x² + 5x - 300) = 0

⇒ x² + 5x - 300 = 0

⇒ x² + 20x - 15x - 300 = 0

⇒ x(x + 20) - 15(x + 20) = 0

⇒ (x - 15)(x + 20) = 0

⇒ x - 15 = 0 or x + 20 = 0

⇒ x = 15 or x = -20.

Since, speed cannot be negative,

∴ x ≠ -20.

∴ x = 15 and x + 5 = 20

Hence, speed of first train = 20 km/hr and second train = 15 km/hr.