Question

At Deoprayag river Alaknanda mixes with the river Bhagirathi and becomes river Ganga. Suppose Alaknanda has a width of 12 m, Bhagirathi has a width of 8 m and Ganga has a width of 16 m. Assume that the depth of water is the same in the three rivers. Let the average speed of water in Alaknanda be 20 km/h and in Bhagirathi be 16 km/h. Find the average speed of the water in the river Ganga.

Answer 1

Let the Depth of each river be h km.

Now, In the Given case, the volume rate of the flow of water will be constant throughout.

This means, Volume rate of the River Alaknanada + Volume Rate of the River Bhagirathi = Volume Rate of the River Ganga.

[This is because, In series connection of Horizontal pipe (here stream of river) the sum of the volume rate equal to the final volume rate flow.]

Volume rate of flow of River Alaknanda = Area of cross section × Velocity of the water in River Alaknanda.

= 12/1000 × h × 20

Similarly,

Volume rate of flow of River Bhagirathi = Area of cross section × Velocity of the water in River Bhagirathi.

= 8/1000 × h × 16

Similarly, Volume rate of flow of River Ganga = Area of cross section × Velocity of the water in River Ganga.

= 16/1000 × h × Velocity

Now, Finally,

16/1000 × h × Velocity =  12/1000 × h × 20 + 8/1000 × h × 16

Cancelling h and 1000 from each side,

16 × Velocity = 12 × 20 + 8 × 16

On Solving, We get, Velocity = 23 km/hr.

Hence, the average velocity of the water in the river Ganga is 23 km/hr.

Hope it helps.

Answer 2

$<b>Answer</b>$

Let the Depth of each river be h km.

$<b>Now, In the Given case, the volume rate of the flow of water will be constant throughout.</b>$

This means, Volume rate of the River Alaknanada + Volume Rate of the River Bhagirathi = Volume Rate of the River Ganga.

[This is because, In series connection of Horizontal pipe (here stream of river) the sum of the volume rate equal to the final volume rate flow.]

$<b>Volume rate of flow of River Alaknanda = Area of cross section × Velocity of the water in River Alaknanda.</b>$

= 12/1000 × h × 20

$<b>Similarly,</b>$

Volume rate of flow of River Bhagirathi = Area of cross section × Velocity of the water in River Bhagirathi.

= 8/1000 × h × 16

$<b>Similarly, Volume rate of flow of River Ganga = Area of cross section × Velocity of the water in River Ganga.</b>$

= 16/1000 × h × Velocity

Now, Finally,

16/1000 × h × Velocity =  12/1000 × h × 20 + 8/1000 × h × 16

$<b>Cancelling h and 1000 from each side,</b>$

16 × Velocity = 12 × 20 + 8 × 16

On Solving, We get, Velocity = 23 km/hr.

$<b>Hence, the average velocity of the water in the river Ganga is 23 km/hr.</b>$

Hope it helps.