BOAT AND STREAM

The Boat and Stream concept involves calculating the speed of a boat (or swimmer) in still water when moving either upstream or downstream in a river or stream. The presence of a stream or river current affects the speed of the boat, and understanding this relationship is crucial for solving problems related to Boat and Stream.

Here are the key concepts:

1. Speed of Boat in Still Water ($B$):

   • This is the speed of the boat when there is no current or stream. It represents the boat's inherent speed and is denoted by $B$ (in km/h or any other unit of speed).

2. Speed of Stream ($S$):

   • This is the speed of the river or stream. It affects the boat's speed when moving upstream or downstream. It is denoted by $S$ (in km/h or any other unit of speed).

3. Speed Downstream ($B+S$):

   • When the boat is moving in the same direction as the stream, the speed downstream is the sum of the boat's speed in still water and the speed of the stream.

4. Speed Upstream ($B-S$):

   • When the boat is moving against the stream, the speed upstream is the difference between the boat's speed in still water and the speed of the stream.

5. Relative Speed:

   • The relative speed of the boat with respect to the stream is the difference between the speed downstream and the speed upstream.

   $\text{Relative Speed}=(\text{Speed Downstream})-(\text{Speed Upstream})$

Speeds:

• Downstream: Combine boat speed and stream speed: Downstream speed = Boat speed + Stream speed
• Upstream: Subtract stream speed from boat speed: Upstream speed = Boat speed - Stream speed

Derived Values:

• Boat speed in still water: Average upstream and downstream speeds: Boat speed = (Upstream speed + Downstream speed) / 2
• Stream speed: Half the difference between downstream and upstream speeds: Stream speed = (Downstream speed - Upstream speed) / 2
• Average speed: Ratio of upstream and downstream speeds multiplied by boat speed: Average speed = (Upstream speed * Downstream speed) / Boat speed

Distance Calculations:

• Still water trip: Use speed and time: Distance = Boat speed * Time
• Downstream and upstream trip (identical distance):
  • Time difference: "t" hours more upstream than downstream.
  • Formula: Distance = (Boat speed^2 - Stream speed^2) * Time difference / (2 * Stream speed)
• Downstream and upstream trip (different distances):
  • Times: t1 for downstream and t2 for upstream.
  • Formula: Boat speed = Stream speed * (Total time / Time difference)