Question

8.
Two trains leave a railway station at the same time. The first train travels towards west and
the second train towards north. The first train travels 5 km/hr faster than the second train.
If after two hours they are 50 km. apart find the average speed of each train.
​

Answer 1

Answer:-

Given:

Two trains leave a station at the same time.

Let the first train be A and second train be B.

And,

Speed of train A is 5 km/h more than that of B.

let the speed of train B be x km/h.

→ Speed of train A = (x + 5) km/h

Distance between the trains after 2 hours = 50 km.

We know that,

Distance = speed × time

→ Distance travelled by train A = speed of train A*2 hours.

[time taken to travel by each train is equal i.e., time taken by train A = time taken by train B ]

→ Distance travelled by train A = (2x) km

Similarly,

→ Distance travelled by train B = 2* (x + 5) = (2x + 10) km.

Now by referring to the image ,

Using Pythagoras Theorem,

→ (Hypotenuse)² = (Opposite side)² + (Adjacent side)²

→ (Distance b/n the trains)² = (Distance travelled by train A)² + (Distance travelled by train B)²

→ (50)² = (2x)² + (2x + 10)²

→ 2500 = 4x² + (2x)² + (10)² + 2 * 2x * 10

[ (a + b)² = a² + b² + 2ab ]

→ 2500 = 4x² + 4x² + 100 + 40x

→ 0 = 8x² + 40x + 100 - 2500

→ 8(x² + 5x - 300) = 0

→ x² + 20x - 15x - 300 = 0

→ x (x + 20) - 15(x + 20) = 0

→ (x - 15) (x + 20) = 0

→ x - 15 = 0

→ x = 15

→ x + 20 = 0

→ x = - 20

Speed can't be negative. So Positive value is taken. i.e., 15 km/h

Hence,

• Total Distance travelled by train A = 2 * (15 + 5) = 40 km.

• total Time taken by train A = 2 hours.

Average speed = Total Distance travelled/total Time taken.

→ Average speed of train A = 40/2

→ Average speed of train A = 20 km/h

• Total Distance travelled by train B = 2 * 15 = 30 km

• Total Time taken by train B = 2 hours.

→ Average speed of train B = 30/2

→ Average speed of train B = 15 km/h.

Therefore, the average speed of train A is 20 km/h and that of train B is 15 km/h.

Answer 2

$\huge \sf \color{purple}{\underline{\underline{Answer}}} :$

Speed of first train is 20 Km/h and speed of 2nd train is 15 Km/h

$\huge \sf \color{purple}{\underline{\underline{Solution}}}:$

➣ Let the 2nd train travel at X km/h

➣Then, the speed of a train is (5 +x) Km/hour.

➣ let the two trains live from station M.

➣ Distance travelled by first train in 2 hours

$\sf \boxed{\colorbox{skyblue}{Distance=speed×time}}$

= MA = 2(x+5) Km.

➣ Distance travelled by second train in 2 hours

= MB = 2x Km

$\sf \color{brown}{By \: Phythagoras \: theorem }$ AB²= MB²+MA²

⟹ 50²=(2(x+5)²+(2x)²

⟹ 2500 = (2x+10)² + 4x²

⟹8x² + 40x - 2400 = 0

⟹x² + 5x - 300 = 0

⟹x² + 20x -15x - 300 = 0

⟹x(x + 20) - 15(x + 20) = 0

⟹ (x + 20)(x -15) = 0

$\sf \boxed{\colorbox{lightgreen}{x=15 \: or \: -20}}$

Taking x = 15 , the speed of second train is 15 Km/h and speed of first train is 20 Km/h