The rate of volume flow of water through a canal is found to be a function of cross -section of canal and velocity of water.show that the rate of volume flow is proportional to the velocity of flow of water
Rateflow=C*area*velocity given
Now, to prove this, look at a sliver of the line, area A, length L. Call that volume deltaV.
Now assume some flow in the canal, so that the speed down the canal is such that the length L passes in t seconds.
The velocity then is L/t. The amount of flow then is Area*L
amountflow= Area*L
amountflow/time=area*L/T
flow rate=area*velocityflow
In the basic equation, the constant C allows for unit conversions.
V=kvA^-2
To show that the rate of volume flow is proportional to the velocity of flow of water, we can begin by considering the definition of volume flow rate.
The volume flow rate (Q) through a canal can be defined as the amount of water (V) passing through a given cross-section of the canal per unit time (t):
Q = V / t
Since the cross-section of the canal (A) and the velocity of water (v) are related, we can express the volume (V) passing through the canal as the product of the cross-sectional area (A) and the distance traveled by the water (d):
V = A * d
By substituting this expression for V into the volume flow rate equation, we get:
Q = (A * d) / t
Next, let's consider the relationship between the distance traveled by water (d) and the velocity of flow (v). The distance traveled can be expressed as the product of the velocity of flow (v) and the time taken (t):
d = v * t
Substituting this expression for d back into the volume flow rate equation, we have:
Q = (A * v * t) / t
Simplifying this equation, we find that the time taken (t) cancels out:
Q = A * v
From this equation, we can see that the volume flow rate (Q) is directly proportional to the velocity of flow (v) when the cross-section of the canal (A) is held constant. Thus, it can be concluded that the rate of volume flow is proportional to the velocity of flow of water.
To show that the rate of volume flow is proportional to the velocity of water flow in a canal, we need to understand the concept of volume flow rate first.
The volume flow rate, also known as volumetric flow rate, is a measure of how much fluid passes through a given point in a specific amount of time. It is represented by the symbol Q and is given by the formula:
Q = A × v
where:
Q = volume flow rate
A = cross-sectional area of the canal
v = velocity of water flow
Now, let's assume that the cross-sectional area of the canal, A, is constant. This means that the width and depth of the canal remain fixed. In such a case, the only factor that can change the volume flow rate, Q, is the velocity of water flow, v.
If the velocity of water flow increases, the volume of water passing through a specific point of the canal will also increase in a given amount of time. This makes intuitive sense, as faster-moving water carries more particles of water per unit time.
Conversely, if the velocity of water flow decreases, the volume of water passing through that specific point of the canal will also decrease in a given amount of time.
Hence, we can conclude that the rate of volume flow is indeed proportional to the velocity of flow of water, ceteris paribus (all other factors remaining constant).