When objects move within a medium that is also moving, like an airplane in the wind or a boat in a river current, understanding relative velocity is crucial. The speed observed by someone on the ground differs from the vehicle’s speedometer reading. For instance, a speed boat is traveling at 100km/hr relative to the water, but its speed relative to a stationary observer on the shore might be different due to the river’s current. This concept highlights that motion is relative to the observer’s frame of reference.
Tailwinds, Headwinds, and Sidewinds: Airplane Examples
Consider an airplane flying with a tailwind, a wind blowing from behind. If the plane’s velocity relative to the air is 100 km/hr and the tailwind’s velocity is 25 km/hr, the plane’s resultant velocity (its velocity relative to the ground) is the sum of these two velocities. In this simple case, the plane travels at 125 km/hr relative to the ground.
The plane is traveling at 125 km/hr relative to the ground due to tailwind.
Conversely, a headwind, blowing against the plane, reduces the resultant velocity. If the same plane encounters a 25 km/hr headwind, its resultant velocity would be 75 km/hr relative to the ground.
Now, imagine a plane traveling South at 100 km/hr encountering a sidewind of 25 km/hr blowing West. Here, the resultant velocity is the vector sum of the two velocities. Since the vectors are at right angles, we use the Pythagorean theorem to find the magnitude of the resultant velocity.
A plane is affected by a sidewind, requiring vector addition to determine resultant velocity.
The magnitude of the resultant velocity is:
(100 km/hr)² + (25 km/hr)² = R²
10,000 km²/hr² + 625 km²/hr² = R²
10,625 km²/hr² = R²
√(10,625 km²/hr²) = R
103.1 km/hr = R
The direction of the resultant velocity can be determined using trigonometry. The angle (θ) between the resultant vector and the southward vector is:
tan(θ) = (opposite/adjacent) = (25/100)
θ = arctan(25/100)
θ = 14.0 degrees
The direction of the resultant is 14.0 degrees West of South, or 256 degrees from due East.
Riverboat Motion: Applying Relative Velocity
The principle of relative velocity also applies to boats in rivers. A motorboat heading straight across a river will be pushed downstream by the current. Even if the boat maintains a speed of 4 m/s directly across the river, its resultant velocity will be greater than 4 m/s and angled downstream.
Suppose a river flows North at 3 m/s and a motorboat travels East at 4 m/s. To find the resultant velocity of the boat relative to an observer on the shore, we again use the Pythagorean theorem:
(4.0 m/s)² + (3.0 m/s)² = R²
16 m²/s² + 9 m²/s² = R²
25 m²/s² = R²
√(25 m²/s²) = R
5.0 m/s = R
The riverboat’s resultant velocity considers both boat speed and river current.
The direction of the resultant velocity is:
tan(θ) = (opposite/adjacent) = (3/4)
θ = arctan(3/4)
θ = 36.9 degrees
The resultant velocity of the boat is 5 m/s at 36.9 degrees North of East.
Riverboat problems typically involve answering three key questions:
- What is the resultant velocity (magnitude and direction) of the boat?
- If the river is X meters wide, how long does it take the boat to cross?
- How far downstream does the boat reach the opposite shore?
We’ve already addressed the first question. The second and third questions require using the average speed equation and careful consideration of the components of motion.
Example Scenario
Let’s say a speed boat is traveling at 100km/hr, East encounters a current traveling 3.0 m/s, North. The river is 80 meters wide.
- What is the resultant velocity of the motorboat?
- How much time does it take the boat to travel shore to shore?
- What distance downstream does the boat reach the opposite shore?
The resultant velocity is 5 m/s at 36.9 degrees (as calculated above).
To find the time to cross the river, we use the average speed equation:
time = distance / average speed
The distance across the river is 80 meters. The average speed in the direction across the river is the boat’s speed relative to the water, which is 4 m/s.
time = (80 m) / (4 m/s) = 20 s
It takes 20 seconds for the boat to cross the river.
During these 20 seconds, the boat also drifts downstream. To find the downstream distance:
distance = average speed * time
The average speed in the downstream direction is the river’s current, which is 3 m/s.
distance = (3 m/s) * (20 s) = 60 m
The boat is carried 60 meters downstream while crossing the river.
Key Takeaway
The motion of the boat can be broken down into two independent components: motion across the river and motion downstream. The boat’s motor determines the speed and distance across the river, while the river current determines the speed and distance downstream. These components act simultaneously and independently.
Consider another example:
A motorboat traveling 4 m/s, East encounters a current traveling 7.0 m/s, North. The river is 80 meters wide.
- What is the resultant velocity of the motorboat?
- How much time does it take the boat to travel shore to shore?
- What distance downstream does the boat reach the opposite shore?
Answers:
a. Resultant velocity: √(4² + 7²) = 8.06 m/s. Direction = arctan(7/4) = 60° North of East.
b. Time to cross: (80 m) / (4 m/s) = 20 s
c. Distance downstream: (7 m/s) * (20 s) = 140 m
An important point to note: the time to cross the river is independent of the current’s velocity. The current only affects how far downstream the boat travels. Perpendicular components of motion are independent.
Check Your Understanding
-
A plane can travel at 80 mi/hr with respect to the air. Determine the resultant velocity (magnitude only) if it encounters a:
a. 10 mi/hr headwind.
b. 10 mi/hr tailwind.
c. 10 mi/hr crosswind.
d. 60 mi/hr crosswind.
Answers:
a. 70 mi/hr
b. 90 mi/hr
c. 80.6 mi/hr
d. 100 mi/hr
-
A motorboat traveling 5 m/s, East encounters a current traveling 2.5 m/s, North.
a. What is the resultant velocity of the motor boat?
b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?
c. What distance downstream does the boat reach the opposite shore?
Answers:
a. 5.59 m/s at 26.6 degrees North of East
b. 16.0 s
c. 40 m
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A motorboat traveling 5 m/s, East encounters a current traveling 2.5 m/s, South.
a. What is the resultant velocity of the motor boat?
b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?
c. What distance downstream does the boat reach the opposite shore?
Answers:
a. 5.59 m/s at 333.4 degrees
b. 16.0 s
c. 40 m
-
A motorboat traveling 6 m/s, East encounters a current traveling 3.8 m/s, South.
a. What is the resultant velocity of the motor boat?
b. If the width of the river is 120 meters wide, then how much time does it take the boat to travel shore to shore?
c. What distance downstream does the boat reach the opposite shore?
Answers:
a. 7.10 m/s at 327.6 degrees
b. 20.0 s
c. 76 m
-
If the current velocity in question #4 were increased to 5 m/s, then
a. how much time would be required to cross the same 120-m wide river?
b. what distance downstream would the boat travel during this time?
Answers:
a. 20.0 s (same as before)
b. 100 m
