Question

A body moving through air at high speed V is found to experience a retarding force F, given be relating F=kAeV^a, where 'a' is the surface area of the body, 'e' the density of air and k is the numerical constant. Use dimensional analysis to deduce the value of 'a'.

Answer 1

Question #78221, Physics / Other

A body moving through air at high speed $V$ is found to experience a retarding force $F$, given by relating $F = kAeV^a$, where 'A' is the surface area of the body, 'e' the density of air and $k$ is the numerical constant. Use dimensional analysis to deduce the value of 'a'.

Solution

$$F = k A e V^ {a};$$ $$[ F ] = k A e V ^ {a} = 1 \times m ^ {2} \times \frac {k g}{m ^ {3}} \times \left(\frac {m}{s}\right) ^ {a} = N = \frac {k g \cdot m}{s ^ {2}};$$ $$m ^ {2} \times \frac {k g}{m ^ {3}} \times \left(\frac {m}{s}\right) ^ {a} = \frac {k g \cdot m}{s ^ {2}};$$ $$\frac {1}{m} \times \left(\frac {m}{s}\right) ^ {a} = \frac {m}{s ^ {2}};$$ $$\frac {m ^ {a - 1}}{s ^ {a}} = \frac {m}{s ^ {2}};$$ $$a = 2$$

Answer: $a = 2$

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