Numerical Methods for Engineers
Numerical Methods for Engineers
7th Edition
ISBN: 9780073397924
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Chapter 1, Problem 21P

As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity. A more fundamental representation of the drag force, which assumes turbulent conditions (i.e., a high Reynolds number), can be formulated as

F d = 1 2 ρ A C d v | v |

where F d = the drag force ( N ) ,   r = fluid density ( kg/m 3 ) , A = the frontal area of the object on a plane perpendicular to the direction of motion ( m 2 ) ,   y = velocity (m/s), and C d = a dimensionless drag coefficient.

(a) Write the pair of differential equations for velocity and position(see Prob. 1.18) to describe the vertical motion of a sphere with diameter d   ( m ) and a density of ρ s ( kg/km 3 ) . The differential equation for velocity should be written as a function of the sphere's diameter.

(b) Use Euler's method with a step size of Δ t = 2 stocompute the position and velocity of a sphere over the first 14 s. Employ the following parameters in your calculation: d = 120 cm, ρ = 1.3  kg/m 3 ,   ρ s = 2700  kg/m 3 , and C d = 0.47 . Assume that the sphere has the initial conditions: x ( 0 ) = 100  m and  v ( 0 ) = 40  m/s .

(c) Develop a plot of your results (i.e., y and v versus t) and use it to graphically estimate when the sphere would hit the ground.

(d) Compute the value for the bulk second-order drag coefficient c d ( kg/m ) . Note that, as described in Prob. 1.3, the bulk second- order drag coefficient is the term in the final differential equation for velocity that multiplies the term v | v | .

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Numerical Methods for Engineers

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