# How to calculate average velocity

## Main Question or Discussion Point

The velocity of a river varies parabolicly with V adjacent to the banks and max velocity d halfway betwee the banks. The flow of river is parallel to the bank.

The width of the river is x.

How do you calculate the average velocity?

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Have you studied calculus yet?

Hi

I have.

This is complicated because the question does not state what type of parabola ie its eqn so I don't know what assumptions can be made. Calculus would only be relevent if the eqn is given.

Unless i am missing something obvious.

Thanks in advance for any help.

what's the water speed next to the banks?

water speed next to banks = V
water speed in center of river = d
speed of river is parabolic with distance across river
distance across river=x
The rest of the question for completeness is given this info what id the angle A needed for the swimmer who swims with speed w to swim directly across to the other side of the bank?

That's strange. Due to friction (viscosity) the speed next to the banks is zero and reaches the maximum value half-way across the river. However, according to your data the situation is just the opposite. Could you please check whether the data you gave in the OP is OK?

elite2
Have you tried to set up the quadratic equation?
I'll try to get you started, lets pick some better symbols:
Let x be your independent variable for position across lake.
Let x=0 be center of lake.
Let v1 represent velocity at river bank.
Let v2 represent velocity at center of river.
Let V(x) be your dependent variable for velocity.

Now come up with a formula for V(x).
Is parabola concave up or down?
What will V(0) be?.
What x value gives you velocity v1 (in terms of V2 and V1).

Last edited:
haruspex
Homework Helper
Gold Member
Due to friction (viscosity) the speed next to the banks is zero and reaches the maximum value half-way across the river. However, according to your data the situation is just the opposite.
Parabolic with x does not necessarily mean the min or max is at x = 0. It was stated that the max is in the middle.

elite 2, do you understand that a parabola means it's a quadratic function of x? V and D should allow you to infer a relationship between the parameters, but not the whole equation exactly. Perhaps that's enough to answer the question.

Btw, the swimmer's best strategy is not to maintain a constant angle, but that's a different question.

K^2
Parabolic with x does not necessarily mean the min or max is at x = 0. It was stated that the max is in the middle.
These are just standard boundary conditions. A real flow will have zero velocity at the banks. But it doesn't matter. If this is just a math problem, you can put any velocity you want.

A parabola is completely determined by 3 values, which you have. Write an equation for a general parabola and set up 3 equations in 3 unknowns for the coefficients. Solve, and you have the equation you can integrate over.

I think i have solved it. The angle is 45 degrees.

The 3 velocities i think are a red herring. Since no actual values are given ie no actual numerical relationship between x and velocity.

Elite2