Answer:
The terminal velocity at sea level is 7.99 m/s
The terminal velocity at an altitude of 5000 m is 10.298 m/s
Explanation:
mass of sphere m = 10 kg
radius of sphere r = 0.5 m
air density at sea level p = 1.22 kg/m^3
drag coefficient Cd = 0.8
terminal velocity = ?
Area of the sphere A = [tex]4\pi r^{2}[/tex] = 4 x 3.142 x [tex]0.5^{2}[/tex] = 3.142 m^2
terminal velocity is gotten from the relationship
[tex]Vt = \sqrt{\frac{2mg}{pACd} }[/tex]
where g = acceleration due to gravity = 9.81 m/s^2
imputing values into the equation
[tex]Vt = \sqrt{\frac{2*10*9.81}{1.22*3.142*0.8} }[/tex] = 7.99 m/s
If at an altitude of 5000 m where air density = 0.736 kg/m^3, then we replace value of air density in the relationship as 0.736 kg/m^3
[tex]Vt = \sqrt{\frac{2mg}{pACd} }[/tex]
[tex]Vt = \sqrt{\frac{2*10*9.81}{0.736*3.142*0.8} }[/tex] = 10.298 m/s
The value of terminal velocity of object is 7.99 m/s and the value of terminal velocity at an altitude of 5000 m is 10.30 m/s.
Given data:
The mass of sphere is, m = 10 kg.
The radius of sphere is, r = 0.5 m.
The density of air is, [tex]\rho = 1.22 \;\rm kg/m^{3}[/tex].
The drag coefficient of object is, [tex]C_{d}=0.8[/tex].
The altitude is, h = 5000 m.
The density of air at altitude is, [tex]\rho' =0.736 \;\rm kg/m^{3}[/tex].
The mathematical expression for the terminal velocity of an object is,
[tex]v_{t}=\sqrt\dfrac{2mg}{\rho \times A \times C_{d}}[/tex]
here,
g is the gravitational acceleration.
A is the area of sphere.
Solving as,
[tex]v_{t}=\sqrt{\dfrac{2 \times 10 \times 9.8}{1.22 \times (4 \pi r^{2}) \times C_{d}}}\\\\\\v_{t}=\sqrt{\dfrac{2 \times 10 \times 9.8}{1.22 \times (4 \pi \times 0.5^{2}) \times 0.8}}\\\\\\\v_{t}=7.99 \;\rm m/s[/tex]
Now, the terminal velocity at the altitude of 5000 m is given as,
[tex]v_{t}'=\sqrt\dfrac{2mg}{\rho' \times A \times C_{d}}[/tex]
Solving as,
[tex]v_{t}'=\sqrt{\dfrac{2 \times 10 \times 9.8}{0.736 \times (4 \pi \times 0.5^{2}) \times 0.8}}\\\\\\v_{t}'=10.30 \;\rm m/s[/tex]
Thus, we can conclude that the value of terminal velocity of object is 7.99 m/s and the value of terminal velocity at an altitude of 5000 m is 10.30 m/s.
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