For objects moving through air near the ground, whose linear size is between a f
ID: 2275713 • Letter: F
Question
For objects moving through air near the ground, whose linear size is between a few millimeters and a few meters, and whose speed is less than a few hundred meters persecond, the drag force is close to a quadratic function of velocity, F(D)=1/2*C(D)A*p*v^2, where p is the density of air near the ground, A is the cross-sectiona larea of the object facing the wind and C(D) is the dimensionless drag coefficient, which deoends on the shape of the object.
a) Verify that the formula is dimensionally correct.
b) A rule of thumb is that in air near the ground where p=1.2 kg/m^3, then F(D)=1/4A*v^2 Show that this formula is consistent with the drag coefficient data for a sphere as it's shown in the graph.
c) estimate the terminal velocity v(T) of a falling skydiver of mass m and cross-sectional area A.
d) Find v(T) in m/s and mile/hr, for a skydiver with A=0.75 m^2 and mass 75 kg.
Explanation / Answer
a) Dim(v) = LT^-1
Dim(A) = L^2
Dim(p) =ML^-3
Dim(C)= 1 {as it is constant}
So, Dim (Apv^2/2) = MLT^-1 = Dim(F)
So,dimensionally it is correct.
b)The figure shows that at low Reynolds numbers, spheres exhibit a roughly constant drag with a CD value ~0.47, and at higher Reynolds numbers they exhibit CD values from 0.1 to 0.2. The decrease in drag occurs for Reynolds numbers greater than ~400,000 indicates an abrupt change from separated laminar to separated turbulent flow. Turbulent flow separates from the sphere's surface later than laminar flow, so the area of low pressure behind the sphere is smaller, reducing net drag. So, the formula becomes F = A*1.2**0.47*v^2/2 = 0.25Av^2(Proved).
c) Let,the terminal velocity be Vt . then, (1/2)*C*p* A*Vt^2 = mg
So, the terminal velocity is Vt = squreroot(2mg/C*p*A) (ans.)
If we assume him as sphere then ,Vt= squreroot(4m*9.8/A) m/s
d)Vt = squreroot(4*75*9.8/0.75) m/s = 62.6 m /s(Ans.)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.