Question

Water flows along a streamline down a river of constant width. Over a short distance, the water slows from speed v to v/3. Which of the following can you correctly conclude about the river's depth?

a. It became deeper by a factor of 3.
b. It became shallower by a factor of 3.
c. It became deeper by a factor of 3^2
d. It became shallower by a factor of 3^2

Answer 1

Answer:

a. It became deeper by a factor of 3.

Explanation:

What we have is water flowing down a river with constant width. The water slows from speed v to v3 over a shirt distance

Using the equation of continuity

A1V1 = A2V2 ----1

A1 is the area of rectangle

V1 is the velocity of water

Area of rectangle = length x width

We rewrite equation 1 as

λ1w1v1 = λ2w2v2

We have w1 = w2

λ1v1 = λ2v2

λ1*v1 = λ2*v/3

λ1 = λ2/3

So it becomes deeper by a factor of 3

Answer 2

The river's depth  became shallower by a factor of $3^2$.

The given parameters;

• width of the river, = w
• initial speed of the river = v
• final speed of the river = v/3

The depth of the river can be obtained by applying third kinematic equation as shown below;

$v_2^2 = v_1^2 + 2gh\\\\h = \frac{v_2^2 - v_1^2}{2g} \\\\h = \frac{(v/3)^2 - v^2}{2g} \\\\h = \frac{v^2 - v^2}{3^2 \times 2g}$

Thus, we can conclude that the river's depth  became shallower by a factor of $3^2$.

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