Question

you plan a trip on which you want to average 90.0km/h. you cover the first half of your distance at an average speed of only 48km/h. what type of vehicle must you use in order to meet your goal? Note that the velocities are based on half the distance not half the time

Answer 1

you plan a trip on which you want to average 90.0km/h. you cover the first half of your distance at an average speed of only 48km/h. what type of vehicle must you use in order to meet your goal? Note that the velocities are based on half the distance not half the time

**Solution:**

$$V_a = 90 \frac{\text{km}}{\text{h}} - \text{average speed};$$ $$V_1 = 48 \frac{\text{km}}{\text{h}} - \text{speed on the first half of the distance};$$ $$d - \text{traveled distance};$$

Formula for the average speed:

$$V_a = \frac{d}{t}$$ $$t = t_1 + t_2 = \frac{\frac{d}{2}}{V_1} + \frac{\frac{d}{2}}{V_2} = \frac{d}{2V_1} + \frac{d}{2V_2} = \frac{d(V_2 + V_1)}{2V_1 V_2}$$

(2)in(1):

$$V_a = \frac{d}{\frac{d(V_2 + V_1)}{2V_1 V_2}} = \frac{2V_1 V_2}{V_2 + V_1}$$ $$2V_1 V_2 = V_2 V_a + V_1 V_a$$ $$V_2 (2V_1 - V_a) = V_1 V_a$$ $$V_2 = \frac{V_1 V_a}{2V_1 - V_a} = \frac{48 \frac{\text{km}}{\text{h}} \cdot 90 \frac{\text{km}}{\text{h}}}{2 \cdot 48 \frac{\text{km}}{\text{h}} - 90 \frac{\text{km}}{\text{h}}} = 720 \frac{\text{km}}{\text{h}}$$

**Answer:** we must use the vehicle with average speed $720 \frac{\text{km}}{\text{h}}$ (airplane).