The ship in the figure below travels along a straight line parallel to the shore and a distance d = 580 m from it. The ship's radio receives simultaneous signals of the same frequency from antennas A and B, separated by a distance L = 700 m apart. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D, which is directly outward from the shore from point B. Determine the wavelength of the radio waves.

Question

The ship in the figure below travels along a straight line parallel to the shore and a distance d = 580 m from it. The ship's radio receives simultaneous signals of the same frequency from antennas A and B, separated by a distance L = 700 m apart. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D, which is directly outward from the shore from point B. Determine the wavelength of the radio waves.

Answer 1

To determine the wavelength of the radio waves, we can use the concept of interference and the given information about the distances between the ship, antennas, and points C and D.

Here's how you can approach the problem:

1. Start by drawing a diagram of the situation. Label the positions of the ship, antennas A and B, point C, and point D. Make sure to include the distances d, L, and the positions of C and D.

2. Since the signals interfere constructively at point C, this implies that the path length from antenna A to point C is equal to the path length from antenna B to point C. In other words, the distances between antenna A and C and between antenna B and C are equal.

3. Use the given distances and the diagram to set up an equation representing the path length equality. Let's call the wavelength of the radio waves λ. The path length from antenna A to point C is given by d + λ/2 (since point C is equidistant from both antennas), and the path length from antenna B to point C is given by L - d + λ/2. Therefore, we have the equation: d + λ/2 = L - d + λ/2.

4. Simplify the equation. Cancel out the λ/2 terms since they appear on both sides of the equation. This leaves us with: d = L - d.

5. Solve the equation for d. Add d to both sides of the equation to get: 2d = L. Rearrange the equation to solve for d: d = L/2.

6. Substitute the given value of L = 700 m into the equation to find the value of d: d = 700/2 = 350 m.

7. Now we have the value of d. To find the wavelength of the radio waves, we can use the equation: λ = 2(d - L/2). Substitute the values of d = 350 m and L = 700 m into the equation to get: λ = 2(350 - 700/2) = 2(350 - 350) = 2(0) = 0 m.

8. The calculated wavelength of 0 m does not make sense in this context, which suggests an error in the information or the approach used. Double-check the given information and ensure that all values are consistent and accurate.