A ship is travelling due east at a speed of 15 km/h.
Find the speed of a boat heading 30° east of north, if
it always appears due north from the ship
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We have a relative velocity problem. A ship travels due east at 15 kilometers per hour. We need to find the speed of a boat that heads 30 degrees east of north, given that the boat always appears due north from the ship's perspective.
Let's set up our coordinate system. We define east as the positive x-axis and north as the positive y-axis. The ship's velocity is 15 kilometers per hour due east, so it's represented as vector (15, 0). The boat heads 30 degrees east of north, which means it makes a 60-degree angle with the positive x-axis. So the boat's velocity components are |V_b| times cosine 60 degrees for x, and |V_b| times sine 60 degrees for y.
Now let's analyze the relative velocity. The relative velocity of the boat with respect to the ship is V_b minus V_s. Since the boat always appears due north from the ship, the relative velocity has zero x-component and some positive y-component. This gives us the equation: (0, |V_b,s|) equals V_b minus V_s.
Now we solve the equation. From the x-component equation, we have zero equals |V_b| times one-half minus 15. Rearranging, |V_b| times one-half equals 15. Therefore, |V_b| equals 15 times 2, which gives us 30 kilometers per hour. So the boat's speed is 30 kilometers per hour.
Let's verify our solution. With the boat speed of 30 kilometers per hour, the boat's velocity components are: x-component equals 30 times cosine 60 degrees, which is 15; y-component equals 30 times sine 60 degrees, which is 26. The relative velocity is (15, 26) minus (15, 0), giving us (0, 26). This confirms that the boat appears due north from the ship. Therefore, our final answer is 30 kilometers per hour.