How to solve river boat problems in relative velocity?

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River Boat Problem || Relative Velocity in 2D || River Man Problem || Motion in a Plane || JEE NEET

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Solving River Boat Problems in Relative Velocity

Introduction

River boat problems are a type of relative velocity problem in which a boat is moving in a river with a current. These problems involve finding the speed and direction of the boat relative to the ground.

Step-by-Step Solution

To solve a river boat problem in relative velocity, follow these steps:

Draw a diagram of the problem, labeling the boat's speed and direction, the speed and direction of the current, and the resulting speed and direction of the boat relative to the ground.

Use the formula: Velocity of boat in still water = (Velocity of boat in river + Velocity of river)

Find the component of the boat's velocity that is perpendicular to the current by using trigonometry, specifically the sine function. This will give you the boat's speed relative to the ground.

Use the Pythagorean theorem to find the boat's resulting speed and direction relative to the ground.

Example

Suppose a boat is traveling at a speed of 12 km/hr in a river that is flowing at a rate of 4 km/hr. What is the boat's speed and direction relative to the ground?

Draw a diagram labeling the boat's speed and direction, the speed and direction of the current, and the resulting speed and direction of the boat relative to the ground.

Use the formula: Velocity of boat in still water = (Velocity of boat in river + Velocity of river)
- Velocity of boat in still water = 12 + 4 = 16 km/hr

Find the component of the boat's velocity that is perpendicular to the current by using trigonometry, specifically the sine function.
- sin θ = Opposite / Hypotenuse
- sin θ = 4 / 16 = 0.25
- θ = sin^-1(0.25) = 14.5°
- The boat's speed relative to the ground is: 16 x sin 14.5° = 4.15 km/hr

Use the Pythagorean theorem to find the boat's resulting speed and direction relative to the ground.
- a² + b² = c²
- a = 12 km/hr
- b = 4.15 km/hr
- c = √(12² + 4.15²) = 12.62 km/hr
- The boat's direction relative to the ground is: tan^-1(4.15 / 12) = 19.4°

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