Vector word problem: resultant velocity | Vectors | Precalculus | Khan Academy | Summary and Q&A

TL;DR

Calculate the speed and direction of a boat after encountering a current using vector addition and trigonometry.

Key Insights

• 🚤 The boat's speed after encountering the current is the magnitude of the resulting vector obtained by adding the boat's speed vector and the current vector.
• 💱 The boat's direction after encountering the current is determined by calculating the inverse tangent of the ratio of the change in Y over the change in X and adjusting the angle to align with convention.
• 🚤 Vector addition and trigonometry are used to calculate the boat's speed and direction after encountering the current.
• 🚤 The boat's original speed and the current are represented as vectors with X and Y components.

Transcript

• [Instructor] We're told a boat is traveling at a speed of 26 kilometers per hour in a direction that is a 300 degree rotation from East. At a certain point it encounters a current at a speed of 15 kilometers per hour in a direction that is a 25 degree rotation from East. Answer two questions about the boat's velocity after it meets the current. A... Read More

Questions & Answers

Q: What is the boat's speed after it meets the current?

The boat's resulting speed, obtained by adding the vectors of the boat's speed and the current, is approximately 31.1 km/h.

Q: What is the direction of the boat's velocity after it meets the current?

The boat's direction is approximately 329 degrees clockwise from East, obtained by calculating the inverse tangent of the ratio of the change in Y over the change in X and adding 360 degrees to align with convention.

Q: How are the boat's speed and direction calculated in this scenario?

The boat's speed and direction are calculated using vector addition and trigonometry. The vectors representing the boat's speed and the current are added by placing the tail of one vector at the head of the other. The resulting vector's magnitude gives the speed, and the inverse tangent of the ratio of the change in Y over the change in X gives the direction.

Q: How is the boat's speed rounded in the calculations?

The boat's speed is rounded to the nearest tenth according to the provided instructions.

Summary & Key Takeaways

• A boat is traveling at a speed of 26 km/h in a direction 300 degrees clockwise from East.

• The boat encounters a current at a speed of 15 km/h in a direction 25 degrees clockwise from East.

• After vector addition, the boat's resulting speed is approximately 31.1 km/h and its direction is approximately 329 degrees clockwise from East.