How to calculate average velocity

[elite 2]

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?

 

[the_emi_guy]

Have you studied calculus yet?

 

[elite 2]

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 relevant if the eqn is given.

Unless i am missing something obvious.

Thanks in advance for any help.

 

[Gordianus]

what's the water speed next to the banks?

 

[elite 2]

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?

 

[Gordianus]

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?

 

[the_emi_guy]

elite2
Have you tried to set up the quadratic equation?
I'll try to get you started, let's 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).

 

[haruspex]

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.

 

[elite 2]

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