Question

Assume Eq. 6-14 gives the drag force on a pilot plus ejection seat just after they are ejected from a plane traveling horizontally at${}^{\mathbf{1300}\mathbf{}\mathbf{km}\mathbf{/}\mathbf{h}}$. Assume also that the mass of the seat is equal to the mass of the pilot and that the drag coefficient is that of a sky diver. Making a reasonable guess of the pilot’s mass and using the appropriate${}^{{\mathbf{v}}_{\mathbf{t}}}$value from Table 6-1, estimate the magnitudes of (a) the drag force on the pilot seatand (b) their horizontal deceleration (in terms of g), both just after ejection. (The result of (a) should indicate an engineering requirement: The seat must include a protective barrier to deflect the initial wind blast away from the pilot’s head)

Short answer

(a) The drag force on the pilot seat is ${}^{2\times {10}^4\mathrm{N}}$

(b) The horizontal deceleration is localid="1654156271401" ${}^{18\mathrm{g}\mathrm{m}/{\mathrm{s}}^2}$

Step-by-step solution

Given

$\mathrm{Speed}\mathrm{of}\mathrm{the}\mathrm{plane}=1300\mathrm{km}/\mathrm{h}$

Determining the concept

This problem is based on the drag force which is a type of friction. This is the force acting opposite to the relative motion of an object moving with respect to the surrounding medium.

Formula:

The terminal speed is given by

$v_t=\sqrt{\frac{2F_g}{CpA}}$

Where C is the drag coefficient,${}^p$is the fluid density, A is the effective cross-sectional area, and${}^{F_g}$is the gravitational force.

(a) Determining the drag force on the pilot seat

From table and equation for the speed of air,

$v_t=\sqrt{\frac{2F_g}{CpA}}$

$CpA=2\frac{mg}{v{t}^2}$

Where,${}^{v_t=60\mathrm{m}/\mathrm{s}}$. Now, estimate the pilot’s mass at about${}^{m=70kg}$

Now, convert,velocity into m/s and plug into equation for drag force

$$\begin{gathered} v=1300\left(1000/3600\right) \\ =360m/s \end{gathered}$$

Thus the drag force is given by,

$$\begin{gathered} D=\frac{1}{2}CpA{v}^2 \\ =\frac{1}{2}\left(2\frac{mg}{{v^2}_t}\right)v^2 \\ =mg{\left(\frac{v}{v_t}\right)}^2 \end{gathered}$$

which yields,

$$\begin{gathered} D=\left(70\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right){\left(360/60\right)}^2 \\ \approx 2\times {10}^4\mathrm{N} \end{gathered}$$

Therefore, the drag force on the pilot seat is${}^{2\times {10}^4\mathrm{N}}$

(b) Determining the horizontal deceleration

Assume the mass of the ejection seat is roughly equal to the mass of the pilot.Thus, Newton’s second law (in the horizontal direction) applied to this system of mass

2m gives the magnitude of acceleration as

$$\begin{gathered} \left|a\right|=\frac{D}{2m} \\ =\frac{g}{2}{\left(\frac{v}{v_t}\right)}^2 \\ =18g\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Therefore, the horizontal deceleration is${}^{18\mathrm{g}\mathrm{m}/{\mathrm{s}}^2}$.