Kevin S. Huang

2014: Problems 12-13

Topic: Dynamics
Concepts: Air resistance

Solution:

Since the helicopter quickly reaches terminal velocity $v_t$ , from kinematics, the time $T$ of flight is given by

T=\frac{h}{v_t} \propto h^1

Thus, $\alpha=1$ so the answer is E.

Topic: Other
Concepts: Dimensional analysis

Solution:

From the previous problem, we found $T=khr^{\beta}\rho^{\delta}W^{\omega}$ . Taking the dimensions of both sides,

[T]=[h][r]^{\beta}[\rho]^{\delta}[W]^{\omega}

Using the fact that $[\rho]=M/L^3$ and $[W]=ML/T^2$ ,

T=L^1L^{\beta}\left(\frac{M}{L^3}\right)^{\delta}\left(\frac{ML}{T^2}\right)^{\omega}

Counting powers of $T$ :

1=-2\omega

\omega=-1/2

Counting powers of $M$ :

0=\delta+\omega

\delta=-\omega=1/2

Counting powers of $L$ :

0=1+\beta-3\delta+\omega

\beta=3\delta-\omega-1=-4\omega-1=2-1=1

Since $\beta=1$ , the answer is D.