## Task 17/61

The passenger aircraft IL 14 P of Deutsche Lufthansa weighs approximately $18000 \mathrm{kp}$ including full payload. It requires about 30 s from the start of rolling until it takes off from the ground and has a speed of around $160 \frac{\mathrm{km}}{\mathrm{h}}$ at the moment of takeoff.

For the following calculations, neglect friction and air resistance and consider the motion of the aircraft as uniformly accelerated.

1. What is the average speed during rolling?
2. What is the rolling distance?
3. What is the acceleration of the aircraft?
4. What force is necessary to produce this acceleration?
5. What work is done by the two engines during rolling for the acceleration?
6. What power (in PS) must each of the two engines provide?
1. We denote by $v_{a}$ the initial velocity, by $v_{e}$ the final velocity, and by $v_{d}$ the average velocity of the aircraft during rolling. Furthermore, let $l$ be the rolling distance and $t$ the required time. It holds that $\frac{v_{a}+v_{e}}{2}=v_{d}$. Given $v_{a}=0 \frac{\mathrm{m}}{\mathrm{s}}, v_{e} \approx 44.4 \frac{\mathrm{m}}{\mathrm{s}}$, we obtain $v_{d} \approx 22.2 \frac{\mathrm{m}}{\mathrm{s}}$.
2. For the rolling distance $l$, it holds that: $l=v_{d} \cdot t \approx 666 \, \text{m} \approx \frac{2}{3} \, \text{km}$.
3. For the acceleration $a$, it holds that: $a=\frac{v_{e}-v_{a}}{t} \approx 1.48 \frac{\mathrm{m}}{\mathrm{s}^{2}}$.
4. According to the fundamental law of dynamics, the force $F$ is calculated as $F=m \cdot a$. For the mass $m$, it holds that $m=\frac{G}{g}$, where $G$ is the weight and $g=9.81 \frac{\mathrm{m}}{\mathrm{s}^{2}}$ is the acceleration due to gravity.

Then, $F=\frac{G \cdot a}{g} \approx 2720 \, \text{kp}$.

5. Work $W=F \cdot l \approx 1810000 \, \text{kpm}$.
6. Power $P=\frac{W}{t} \approx 60000 \, \frac{\text{kpm}}{\text{s}}$. This corresponds to 800 HP, so that each engine must deliver about 400 HP.