Question

Two streams merge to form a river. One stream has a width of 8.3 m, depth of 3.2 m, and current speed of 2.2 m/s. The other stream is 6.8 m wide and 3.2 m deep, and flows at 2.4 m/s. If the river has width 10.4 m and speed 2.8 m/s, what is its depth?

Answer 1

The depth of the resulting stream is 3.8 meters.

Step-by-step explanation:

Under the assumption that streams are formed by incompressible fluids, so that volume flow can observed conservation:

`\dot V_(1) + \dot V_(2) = \dot V_(3)` (1)

All volume flows are measured in cubic meters per second.

Dimensionally speaking, we can determine the depth of the resulting stream (
`h_(3)`), in meters, by expanding (1) in this manner:

`w_(1)\cdot h_(1)\cdot v_(1) + w_(2)\cdot h_(2)\cdot v_(2) = w_(3)\cdot h_(3)\cdot v_(3)`

`h_(3) = (w_(1)\cdot h_(1)\cdot v_(1)+w_(2)\cdot h_(2)\cdot v_(2))/(w_(3)\cdot v_(3))` (2)

`v_(1), v_(2)` - Speed of the merging streams, in meters per second.

`h_(1), h_(2)` - Depth of the merging streams, in meters.

`w_(1), w_(2)` - Width of the merging streams, in meters.

`w_(3)` - Width of the resulting stream, in meters.

`v_(3)` - Speed of the resulting stream, in meters per second.

If we know that
`w_(1) = 8.3\,m`,
`h_(1) = 3.2\,m`,
`v_(1) = 2.2\,(m)/(s)`,
`w_(2) = 6.8\,m`,
`h_(2) = 3.2\,m`,
`v_(2) = 2.4\,(m)/(s)`,
`w_(3) = 10.4\,m` and
`v_(3) = 2.8\,(m)/(s)`, then the depth of the resulting stream is:

`h_(3) = ((8.3\,m)\cdot (3.2\,m)\cdot \left(2.2\,(m)/(s) \right) + (6.8\,m)\cdot (3.2\,m)\cdot \left(2.4\,(m)/(s) \right))/((10.4\,m)\cdot \left(2.8\,(m)/(s) \right))`

`h_(3) = 3.8\,m`

The depth of the resulting stream is 3.8 meters.