Vector Resultant: 10 km/hr car + 2 km/hr wind = resultant ??? Draw vectors head-to-tail and then add using Pythagoras' theorem.
Understand the Problem
The question involves calculating the resultant vector from two given vectors: one from a car moving at 10 km/hr and the other from wind at 2 km/hr. The approach includes using the Pythagorean theorem to find the resultant magnitude and the angle to determine the direction.
Answer
The resultant vector is approximately $10.198 \, \text{km/hr}$ in a direction $E11.31^\circ N$.
Answer for screen readers
The resultant vector is approximately $10.198 , \text{km/hr}$ in a direction $E11.31^\circ N$.
Steps to Solve
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Identify the vectors The car's velocity vector is 10 km/hr (horizontal, East direction). The wind's velocity vector is 2 km/hr (vertical, North direction).
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Set up the Pythagorean theorem To find the resultant vector (R), set up the equation using the Pythagorean theorem: $$ R^2 = 10^2 + 2^2 $$
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Calculate the magnitude of the resultant vector Substituting values into the equation: $$ R = \sqrt{10^2 + 2^2} $$ Calculate: $$ R = \sqrt{100 + 4} = \sqrt{104} \approx 10.198 \text{ km/hr} $$
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Find the angle of the resultant vector Use the tangent function to find the angle ($\theta$): $$ \tan \theta = \frac{2}{10} $$ Then calculate $\theta$: $$ \theta = \tan^{-1}\left(\frac{2}{10}\right) $$
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Calculate the angle Using a calculator: $$ \theta \approx 11.31^\circ $$
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Summarize the resultant vector The resultant vector is approximately 10.198 km/hr at an angle of 11.31° North of East.
The resultant vector is approximately $10.198 , \text{km/hr}$ in a direction $E11.31^\circ N$.
More Information
The magnitude and direction of the resultant vector illustrate how two separate vectors combine to create a new vector that represents the overall effect of both velocities (car and wind). Understanding vector addition is crucial in fields such as physics, engineering, and navigation.
Tips
- Confusing the direction of the vectors when drawing them.
- Incorrectly applying the Pythagorean theorem (e.g., missing squares).
- Not calculating the angle using the proper trigonometric function (tangent should be used for opposite/adjacent).
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