To tackle the problem of the two boats, B1 and B2, moving towards a bank with a water flow velocity of 10 m/s away from the bank, we need to analyze their velocities from different reference frames. This will help us understand how the direction of velocity can change based on the observer's point of view.
Understanding the Scenario
Initially, both boats are 100 m from the bank. After 1 second, B1 is 50 m away, and B2 is 25 m away. This means both boats are moving towards the bank, but we also need to consider the effect of the water current flowing away from the bank.
Defining the Reference Frames
We can analyze the velocities from three different reference frames:
- The bank (stationary reference frame)
- The water (moving reference frame)
- The boats themselves (moving reference frame)
1. Velocity from the Bank's Reference Frame
In this frame, the bank is stationary, and we need to account for the water's flow velocity. The velocities of the boats can be calculated as follows:
- For B1:
- Initial distance from the bank = 100 m
- Distance after 1 second = 50 m
- Distance traveled = 100 m - 50 m = 50 m
- Velocity of B1 = Distance traveled / Time = 50 m / 1 s = 50 m/s towards the bank
- Net velocity of B1 = 50 m/s (towards the bank) - 10 m/s (away from the bank) = 40 m/s towards the bank
- For B2:
- Initial distance from the bank = 100 m
- Distance after 1 second = 25 m
- Distance traveled = 100 m - 25 m = 75 m
- Velocity of B2 = Distance traveled / Time = 75 m / 1 s = 75 m/s towards the bank
- Net velocity of B2 = 75 m/s (towards the bank) - 10 m/s (away from the bank) = 65 m/s towards the bank
2. Velocity from the Water's Reference Frame
In this frame, the water is stationary, and we need to consider the velocities of the boats relative to the water:
- For B1:
- Velocity of B1 relative to water = 50 m/s (towards the bank)
- Water flow velocity = 10 m/s (away from the bank)
- Net velocity of B1 = 50 m/s (towards the bank) + 10 m/s (away from the bank) = 60 m/s towards the bank
- For B2:
- Velocity of B2 relative to water = 75 m/s (towards the bank)
- Water flow velocity = 10 m/s (away from the bank)
- Net velocity of B2 = 75 m/s (towards the bank) + 10 m/s (away from the bank) = 85 m/s towards the bank
3. Velocity from the Boats' Reference Frame
In this frame, we consider the boats as stationary. The velocities of the boats relative to each other and the water will be different:
- For B1:
- Relative to B1, B2 is moving towards the bank at a speed of 25 m/s (since B1 is stationary in this frame).
- Water flow is still 10 m/s away from the bank, so B1 perceives the water moving away at that speed.
- For B2:
- Relative to B2, B1 is moving towards the bank at a speed of 25 m/s.
- Water flow is still 10 m/s away from the bank, so B2 perceives the water moving away at that speed.
Direction of Velocity Explained
The direction of velocity can be confusing because it depends on the chosen reference frame. Here’s a breakdown:
- In the bank's frame, velocities towards the bank are positive, while the water's flow away from the bank is negative.
- In the water's frame, the boats' velocities are positive towards the bank, and the water's flow is considered stationary.
- In the boats' frame, the boats are stationary, and the water's flow is perceived as moving away from them.
Understanding these different perspectives helps clarify how motion is relative and how velocities can change based on the observer's frame of reference. Each frame provides valuable insights into the motion of the boats and the effect of the water current.
