Consider a two-dimensional incompressible velocity potential phi = cos theta + theta, where B and L are constants. Locate the stagnation point(s) in this flow field. Also locate the point(s) where the pressure in the flow field will be maximum.

solution:

Increasing the magnification and decreasing the field view

we are given:

a(t)=4*t^2-)

where t= time in seconds, and a(t) = acceleration as a function of time.

and

x(0)=-2 )

x(2) = -20 )

where x(t) = distance travelled as a function of time.

need to find x(4).

from (1), we express x(t) by integrating, twice.

velocity = v(t) = integral of (1) with respect to t

v(t) = 4t^3/3 - 2t + k1     )

where k1 is a constant, to be determined.

integrate (4) to find the displacement x(t) = integral of (4).

x(t) = integral of v(t) with respect to t

= (t^4)/3 - t^2 + (k1)t + k2   )   where k2 is another constant to be determined.

from (2) and (3)

we set up a system of two equations, with k1 and k2 as unknowns.

x(0) = 0 - 0 + 0 + k2 = -2   => k2 = 2   )

substitute (6) in (3)

x(2) = (2^4)/3 - (2^2) + k1(2) -2   = -20

16/3 -4 + 2k1 -2 = -20

2k1 = -20-16/3 +4 +2 = -58/3

=>

k1 = -29/3   )

thus substituting (6) and (7) in (5), we get

x(t) = (t^4)/3 - t^2 - 29t/3 + 2   )

which, by putting t=4 in (8)

x(4) = (4^4)/3 - (4^2 - 29*4/3 +2

= 86/3, or

= 28 2/3, or

= 28.67 (to two places of decimal)

Smith put forth the notion of the invisible hand in arguing that free individuals operating in a free economy, making decisions that are primarily focused on their self-interest logically take actions that benefit society as a whole, even though such beneficial results were not the specific focus or intent of those ...

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