You are fishing in a boat anchored in a bay. You notice that the current is in the eastern direction with small wave crests that are 15 meters apart, and they are gently rocking the boat once every 5 seconds. You decide to lift the anchor and begin trolling slowly to the west at 2.p meters/second. Explain the strategy for calculating how often the wave crests will rock the boat in this situation based on your investigation
To calculate how often the wave crests will rock the boat given the current and the boat's speed, you will need to determine the time it takes for the boat to cover a distance equal to the distance between wave crests.
1. Calculate the speed of the current:
- From the information provided, you notice that the wave crests are 15 meters apart.
- Given that the wave crests are rocking the boat once every 5 seconds, you can determine the speed of the current using the formula: speed = distance/time.
- Therefore, speed of the current = 15 meters / 5 seconds = 3 meters/second (m/s).
2. Consider the direction of the current and boat speed:
- The current is in the eastern direction, while the boat is trolling to the west at 2.p meters/second.
- Since the boat is traveling opposite to the eastward current, the effective speed of the boat will be the difference between the boat speed and the current speed.
- Effective boat speed = boat speed - current speed = 2.p m/s - 3 m/s = -0.3 m/s.
3. Determine the time it takes for the boat to cover a distance equal to the wave crest spacing:
- The time interval between wave crests rocking the boat is provided as 5 seconds.
- The distance traveled by the boat during this time is equal to the effective boat speed multiplied by the time interval: distance = speed × time.
- Therefore, distance = -0.3 m/s × 5 s = -1.5 meters.
4. Calculate the frequency of wave crests rocking the boat:
- Since the distance between wave crests is 15 meters and the boat covers -1.5 meters during the time it takes for a wave crest to rock the boat (5 seconds), we can divide the wave crest spacing by the distance covered per time interval to find the frequency.
- Frequency = wave crest spacing / distance covered per time interval = 15 meters / -1.5 meters = -10.
5. Interpretation:
- The calculated frequency of -10 indicates that the wave crests will rock the boat once every 10 seconds with the given boat speed and current direction.
- Note that the negative frequency simply indicates that the wave crests and boat motion are in opposite directions.
To calculate how often the wave crests will rock the boat in this situation, we need to consider the effects of the current and the speed at which the boat is trolling.
Here is the strategy for calculating the rocking frequency:
1. Determine the speed of the current: You mentioned that the current is flowing in the eastern direction. Let's assume that the speed of the current is C meters/second.
2. Calculate the relative speed between the boat and the waves: Since the boat is trolling to the west at a speed of 2.p meters/second, we need to consider the relative speed between the boat and the waves. The relative speed, R, can be calculated by subtracting the speed of the boat from the speed of the waves. In this case, R = C - 2.p.
3. Calculate the time it takes for the boat to move between two consecutive wave crests: Since the wave crests are 15 meters apart, we need to determine the time it takes for the boat to cover this distance at the relative speed R. The time, T, can be calculated using the formula T = D / R, where D is the distance between two consecutive wave crests.
4. Determine the rocking frequency: The rocking frequency is the reciprocal of the time it takes for the boat to move between two consecutive wave crests. So, the rocking frequency, F, can be calculated as F = 1 / T.
By following these steps, you can calculate how often the wave crests will rock the boat in this situation based on the investigation.