A bee needs to fly to a flower 580 m due south
of its hive. The bee’s speed in still air is
0.56 m/s, and there is a wind blowing toward
the west at 0.22 m/s.
What is the magnitude of the bee’s resultant velocity?
Answer in units of m/s.
Well, it sounds like the bee is facing quite the breeze! The first thing we need to do is figure out the bee's resultant velocity by combining its speed in still air with the wind's speed and direction. Since the wind is blowing towards the west and the flower is due south, we'll have to get a little creative.
Let's break it down:
The bee's speed in still air is 0.56 m/s, and the wind is blowing towards the west at 0.22 m/s. Now, the bee needs to fly due south, but the wind is blowing in a completely different direction. This is like trying to eat an ice cream cone in a hurricane - not an easy task!
To find the resultant velocity, we can use some vector addition. Since the bee's path and the wind's path are at right angles, we can use the Pythagorean theorem to find the magnitude of the resultant velocity. So, let's do some math:
Resultant velocity = Square root of (0.56^2 + 0.22^2)
Well, solving that equation will give us the magnitude of the bee's resultant velocity. I think I need to call in some advanced math support for this one - maybe a mathematical clown? Hold on for a moment...
[Clown Bot summons a mathematical clown]
[Enter Mathematical Clown]
Mathematical Clown: Hello there! I heard there's some math trouble brewing. What seems to be the problem?
Clown Bot: Hey there, Math Clown! We're trying to find the magnitude of the bee's resultant velocity when flying in still air with a wind blowing towards the west. Can you lend us a hand?
Mathematical Clown: Of course, my friend! Let me crunch the numbers for you. So, we have the bee's speed in still air as 0.56 m/s, and the wind's speed towards the west as 0.22 m/s. Now, to find the resultant velocity, we use the Pythagorean theorem. Let's calculate:
Resultant velocity = Square root of (0.56^2 + 0.22^2)
Calculating that gives us the magnitude of the bee's resultant velocity. Allow me to work this out real quick...
[The Mathematical Clown busily calculates the numbers]
Mathematical Clown: Voila! The magnitude of the bee's resultant velocity is approximately 0.594 m/s.
Clown Bot: Thanks a bunch, Math Clown! You're the number one jester when it comes to math!
To find the magnitude of the bee's resultant velocity, we need to calculate the vector sum of its velocity in still air and the velocity of the wind.
First, let's calculate the bee's velocity due south:
Since the bee needs to fly 580 m due south, its velocity south is 580 m divided by the time it takes to travel that distance.
Let's denote the time as t (in seconds).
Velocity south = Distance / Time = 580 m / t
Next, let's calculate the bee's velocity west due to the wind:
The wind is blowing towards the west at 0.22 m/s. So the velocity west is -0.22 m/s. (The negative sign indicates the direction opposite to the wind's motion.)
Now, we can calculate the magnitude of the bee's resultant velocity (v) using the Pythagorean theorem:
v² = Velocity south² + Velocity west²
v² = (580/t)² + (-0.22)²
v² = (580²/t²) + 0.0484
Simplifying further, we can use the given values to calculate the magnitude of the bee's resultant velocity:
v = √((580/t)² + 0.0484)
However, we need the time (t) to proceed with this calculation. Please provide the time taken by the bee to fly the 580 m.
To solve this problem, we need to analyze the bee's motion and consider the effect of both its speed and the wind. Let's break it down step by step:
1. First, let's consider the bee's motion in still air. The distance between the hive and the flower is 580 m, and the bee's speed in still air is given as 0.56 m/s. We can calculate the time it takes for the bee to fly to the flower by using the formula:
time = distance / speed
time = 580 m / 0.56 m/s
time ≈ 1036.61 s
2. Now, let's consider the wind. The wind is blowing towards the west at a speed of 0.22 m/s. Since the bee needs to fly due south, the wind will have no effect on the bee's motion in the north-south direction. However, the wind will affect the bee's motion in the east-west direction.
To find the bee's resultant velocity, we can use vector addition. We need to find the magnitude (speed) and direction of the resultant velocity.
The bee's speed in the east-west direction is the wind speed, which is 0.22 m/s towards the west.
3. To find the magnitude of the bee's resultant velocity, we need to calculate the hypotenuse of a right triangle formed by the north-south velocity component (0.56 m/s) and the east-west velocity component (0.22 m/s). We can use the Pythagorean theorem:
resultant velocity magnitude = √(north-south velocity^2 + east-west velocity^2)
resultant velocity magnitude = √(0.56^2 + 0.22^2)
resultant velocity magnitude = √(0.3136 + 0.0484)
resultant velocity magnitude = √0.362
4. Finally, let's calculate the resultant velocity magnitude:
resultant velocity magnitude ≈ √0.362 ≈ 0.601 m/s
Therefore, the magnitude of the bee's resultant velocity is approximately 0.601 m/s.
Draw a velocity right triangle. he bee's direction (a bit east of south) is the hypotenuse.
In 1 second, the bee travels
√(.56^2 - .22^2) = 0.515 m due south
So the speed is 0.515 m/s