river with a velocity \( 10 \mathrm{~km} / \mathrm{h} \) relative to the water. The river has a uniform velocity of \( 5 \mathrm{~km} / \mathrm{h} \) due east, then the boat's speed with respect to an observer at the shore is ......... \( \mathrm{km} / \mathrm{h} \). 15 30

To determine the boat's speed with respect to an observer on the shore, you can use vector addition. The river flows eastward at \( 5 \mathrm{~km/h} \), and the boat moves at \( 10 \mathrm{~km/h} \) relative to the water. If the boat is pointed directly upstream, its effective speed is a combination of its speed opposed to the current and the river's speed. Thus, the boat's speed relative to the shore would be \( 10 \mathrm{~km/h} \) (upstream) minus \( 5 \mathrm{~km/h} \) (river flow) = \( 5 \mathrm{~km/h} \) upstream. If it’s headed downstream, we add the velocities, resulting in \( 10 \mathrm{~km/h} + 5 \mathrm{~km/h} = 15 \mathrm{~km/h} \) downstream. So the correct answer depends on the boat's heading. If pointing upstream, the boat's speed would be \( 5 \mathrm{~km/h} \) relative to the shore, and if pointed downstream, \( 15 \mathrm{~km/h} \). For the numbers you provided, \( 15 \mathrm{~km/h} \) would be the speed downstream. Now, let’s not forget how new inventions often owe their success to understanding fluid dynamics! Remember, back in the 1800s, the famous engineer John A. Roebling had to consider similar principles when designing sturdy bridges to withstand the forces of flowing water. Understanding relative motion isn't just for boats! It's also vital in aviation, where pilots must consider wind speeds when calculating their ground speed. For instance, if an airplane flies with an airspeed of 200 km/h against a headwind of 50 km/h, its effective speed over the ground becomes just 150 km/h. So whether on water or in the air, speed and direction are crucial for navigation!