Verification of the Relation V = KAtu

Explanation:
To verify the correctness of the relation V = KAtu, we need to understand the physical quantities involved and their respective dimensions. Let's break down the relation and analyze each term.

Volume (V):
Volume is a physical quantity that describes the amount of space occupied by an object or substance. The dimension of volume is given by [L^3] (length cubed).

Cross-sectional area (A):
Cross-sectional area is the area of the shape formed when a solid is cut perpendicular to its length. The dimension of area is given by [L^2] (length squared).

Velocity (u):
Velocity is a physical quantity that describes the rate at which an object changes its position. The dimension of velocity is given by [LT^-1] (length per unit time).

Constant (K):
The constant (K) is dimensionless, meaning it does not have any physical units associated with it. It is a proportionality constant that relates the volume, cross-sectional area, and velocity of the water.

Verification:
To verify the relation V = KAtu, we can analyze the dimensions of each term:

The dimension of V is [L^3].
The dimension of A is [L^2].
The dimension of u is [LT^-1].
The dimension of K is dimensionless.

Now, let's check if the dimensions of both sides of the equation are consistent:

LHS (Left Hand Side): V = [L^3]
RHS (Right Hand Side): KAtu = [L^2] * [LT^-1] * [LT^-1] = [L^3]

Since the dimensions on both sides of the equation match, we can conclude that the relation V = KAtu is correct.

Conclusion:
The relation V = KAtu is verified to be correct based on the analysis of the dimensions of each term involved. This relation shows how the volume of water passing through a point in a canal is related to the cross-sectional area of the canal and the velocity of the water. The constant K represents the proportionality between these quantities.

Verification of the relation:
To verify the correctness of the relation V = KAtu, we need to analyze the dimensions of each term and check if they are consistent.

Dimensions of the terms:
- Volume V: The dimension of volume is [L^3], where L represents length.
- Cross section A: The dimension of cross section is [L^2], as it represents an area.
- Velocity u: The dimension of velocity is [LT^-1], where T represents time.
- Constant K: The dimension of a dimensionless constant is 1.

Dimensions of the relation:
The relation V = KAtu can be rewritten as V = KtuA.
Analyzing the dimensions of each term:
- LHS (left-hand side): V has the dimension [L^3].
- RHS (right-hand side): KtuA has the dimension [LT^-1 * T * L^2] = [L^3T^-1].

Comparing the dimensions:
The dimensions on both sides of the equation are [L^3], indicating that they are consistent. Therefore, the relation V = KAtu is correct.

Explanation:
The relation V = KAtu represents the volume of water passing through any point in a canal during time t. It can be derived by considering the factors affecting the volume of water flow.

- Cross section A: The cross section of the canal represents the area through which the water flows. A larger cross section allows more water to pass through.
- Velocity u: The velocity of water determines how fast it flows through the canal. A greater velocity means more water passes through in a given time.
- Time t: The duration of time determines the total volume of water passing through.

The constant K in the relation is a dimensionless constant that accounts for other factors such as the shape of the canal, roughness of the surface, and other hydraulic characteristics. It is a proportionality constant that ensures the relation holds true.

By multiplying the cross section A, velocity u, and time t, we obtain the volume of water passing through any point in the canal. The constant K accounts for the specific characteristics of the canal and ensures the relation is correct.

In conclusion, the relation V = KAtu is a valid representation of the volume of water passing through any point in the canal during a given time. The dimensions of each term in the relation are consistent, indicating its correctness.