When we observe motion, it’s crucial to understand that it’s relative. Imagine a plane flying in the sky or A Boat Is Traveling East Across A River. The speed we perceive isn’t just the plane’s engine or the boat’s motor at work. External factors like wind or river currents play a significant role. This concept of motion relative to an observer is key to understanding these scenarios. Let’s delve into how these factors influence the actual velocity of moving objects.
The Impact of Wind on Airplanes
Consider an airplane. If there’s a tailwind (wind blowing from behind), the plane’s speed increases relative to the ground. If the plane is traveling at 100 km/hr relative to the air, and a 25 km/hr tailwind is present, the resulting velocity is 125 km/hr. It’s a straightforward addition.
An illustration showing the effect of tailwind on the velocity of the airplane
However, a headwind (wind blowing from the front) decreases the plane’s resulting velocity. In the same scenario, a 25 km/hr headwind would reduce the plane’s ground speed to 75 km/hr.
Things get more interesting with a side wind. If our plane is heading South at 100 km/hr and encounters a Westward side wind of 25 km/hr, we need to use vector addition to find the resultant velocity. This involves the Pythagorean theorem and trigonometric functions to determine both the magnitude and direction of the plane’s actual movement.
The diagram showing the vector addition of side wind on the airplane’s velocity
Here’s how to calculate it:
- Magnitude: √(100² + 25²) = 103.1 km/hr
- Direction: tan⁻¹(25/100) = 14.0 degrees West of South (or 256 degrees from East)
Riverboat Problems: Navigating the Current
Just like wind affects airplanes, a river current affects a boat is traveling east across a river. If a motorboat aims straight across a river, it won’t land directly opposite its starting point because the current will push it downstream. The boat might be moving at 4 m/s across the river, but its actual speed and direction relative to an observer on the shore will be different.
The resultant velocity is again a vector sum – the boat’s velocity plus the river’s velocity. If the river flows North at 3 m/s and the boat heads East at 4 m/s, the resultant velocity is:
- Magnitude: √(4² + 3²) = 5 m/s
- Direction: tan⁻¹(3/4) = 36.9 degrees North of East
A visual representation of the vector addition for a boat crossing a river, showcasing the boat’s velocity and the river’s current.
Riverboat problems often involve these questions:
- What is the resultant velocity of the boat?
- How long does it take to cross the river if it’s a certain width?
- How far downstream does the boat land on the opposite bank?
Solving Riverboat Problems: An Example
Let’s consider a detailed example:
Example: A motorboat travels East at 4 m/s across a river with a Northward current of 3 m/s. The river is 80 meters wide.
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Resultant Velocity: We already know this is 5 m/s at 36.9 degrees North of East.
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Time to Cross: This depends only on the boat’s velocity across the river.
- Time = Distance / Speed = 80 meters / 4 m/s = 20 seconds
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Downstream Distance: This depends on the river’s current and the time spent crossing.
- Distance = Speed Time = 3 m/s 20 seconds = 60 meters
The key here is to understand that the boat’s motion across the river and its motion downstream are independent. The time to cross depends only on the boat’s speed perpendicular to the current, and the downstream distance depends only on the current’s speed and that crossing time.
Another Example: Varying the Current
Consider a similar scenario: A motorboat travels East at 4 m/s across a river 80 meters wide, but now the current is a stronger 7 m/s North.
- Resultant Velocity: √(4² + 7²) = 8.06 m/s at 60 degrees North of East.
- Time to Cross: Still 20 seconds! (80 meters / 4 m/s). The time to cross the river remains the same even with a stronger current.
- Downstream Distance: 7 m/s * 20 seconds = 140 meters.
This highlights a critical point: The speed of the current does not affect the crossing time. It only affects how far downstream the boat ends up. The components of motion that are perpendicular to each other are independent.
Check Your Understanding
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An airplane flies at 80 mi/hr. Find its resultant speed with:
a) A 10 mi/hr headwind: 70 mi/hr
b) A 10 mi/hr tailwind: 90 mi/hr
c) A 10 mi/hr crosswind: 80.6 mi/hr
d) A 60 mi/hr crosswind: 100 mi/hr
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A boat moves East at 5 m/s in a river with a 2.5 m/s North current.
a) Resultant velocity: 5.59 m/s at 26.6 degrees North of East
b) Time to cross an 80m wide river: 16.0 s
c) Distance downstream: 40 m
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A boat moves East at 5 m/s in a river with a 2.5 m/s South current.
a) Resultant velocity: 5.59 m/s at 333.4 degrees
b) Time to cross an 80m wide river: 16.0 s
c) Distance downstream: 40 m
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A boat moves East at 6 m/s in a river with a 3.8 m/s South current.
a) Resultant velocity: 7.10 m/s at 327.6 degrees
b) Time to cross a 120m wide river: 20.0 s
c) Distance downstream: 76 m
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If the current in #4 increases to 5 m/s:
a) Time to cross: 20.0 s (unchanged)
b) Distance downstream: 100 m
Understanding relative velocity, especially when a boat is traveling east across a river, involves breaking down motion into independent components. This allows us to accurately predict the resulting velocity, crossing time, and downstream displacement. It’s a fundamental concept in physics with practical applications in navigation and more.
