A plane can travel with a speed of 80 mi/hr with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a:
10 mi/hr cross wind;
60 mi/hr cross wind;
10 mi/hr tailwind;
10 mi/hr headwind.
I have no idea what equation to use to solve each of these?
Well, determining the resultant velocity of the plane is a piece of cake! We can use a little bit of vector addition to find our answers.
For each scenario:
1. 10 mi/hr crosswind: Imagine the plane is moving with 80 mi/hr forward, and the crosswind is pushing it sideways with 10 mi/hr. To find the resultant velocity, we can use the Pythagorean theorem:
Resultant velocity = √(80^2 + 10^2) = √(6400 + 100) = √6500 ≈ 80.62 mi/hr.
2. 60 mi/hr crosswind: In this case, the crosswind is quite strong. Using the same approach as before:
Resultant velocity = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 mi/hr (Wow, that's a fast plane! Can it beat Usain Bolt?)
3. 10 mi/hr tailwind: When there's a tailwind, the wind is pushing the plane in the same direction it's moving. So, we can simply add the velocities:
Resultant velocity = 80 mi/hr + 10 mi/hr = 90 mi/hr (That's what I call a helping wind!)
4. 10 mi/hr headwind: On the other hand, when there's a headwind, it's pushing against the plane's motion. In this case:
Resultant velocity = 80 mi/hr - 10 mi/hr = 70 mi/hr (A bit slower, but still flying high!)
And there you have it, the magnitudes of the resultant velocities for each scenario. Just remember, when it comes to wind, sometimes it's a friend, and sometimes it blows your hair in all the wrong directions!
To solve this problem, we can use vector addition to determine the resultant velocity of the plane.
1. Crosswind of 10 mi/hr:
When encountering a crosswind, we can use the Pythagorean theorem to determine the magnitude of the resultant velocity.
Resultant velocity = √(80^2 + 10^2) = √(6400 + 100) = √6500 ≈ 80.62 mi/hr
2. Crosswind of 60 mi/hr:
Similarly, when encountering a stronger crosswind, we can use the Pythagorean theorem to determine the magnitude of the resultant velocity.
Resultant velocity = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 mi/hr
3. Tailwind of 10 mi/hr:
When encountering a tailwind, we can simply add the speeds of the plane and the wind.
Resultant velocity = 80 + 10 = 90 mi/hr
4. Headwind of 10 mi/hr:
When encountering a headwind, we can subtract the speed of the wind from the speed of the plane.
Resultant velocity = 80 - 10 = 70 mi/hr
So, the resultant velocities in each scenario are:
- For a crosswind of 10 mi/hr: 80.62 mi/hr
- For a crosswind of 60 mi/hr: 100 mi/hr
- For a tailwind of 10 mi/hr: 90 mi/hr
- For a headwind of 10 mi/hr: 70 mi/hr
To determine the resultant velocity of the plane in each scenario, you need to use vector addition. The velocity of the plane with respect to the ground is the combination of its velocity with respect to the air and the velocity of the wind.
First, let's define some terms:
- The speed of the plane with respect to the air is given as 80 mi/hr.
- A crosswind is a wind blowing perpendicular to the direction of the plane's motion.
- A tailwind is a wind blowing in the same direction as the plane's motion.
- A headwind is a wind blowing opposite to the direction of the plane's motion.
Now, let's solve each scenario:
1. 10 mi/hr crosswind:
To find the resultant velocity, use the Pythagorean theorem:
Resultant velocity = √(80^2 + 10^2) = √(6400 + 100) = √6500 ≈ 80.62 mi/hr
2. 60 mi/hr crosswind:
Again, use the Pythagorean theorem:
Resultant velocity = √(80^2 + 60^2) = √(6400 + 3600) = √10000 = 100 mi/hr
3. 10 mi/hr tailwind:
In this case, the wind supports the plane's motion, so the resultant velocity is the sum of the plane's speed and the tailwind:
Resultant velocity = 80 mi/hr + 10 mi/hr = 90 mi/hr
4. 10 mi/hr headwind:
Here, the wind opposes the plane's motion, so the resultant velocity is the difference between the plane's speed and the headwind:
Resultant velocity = 80 mi/hr - 10 mi/hr = 70 mi/hr
By using the Pythagorean theorem to find the magnitude of the resultant velocity and considering the direction of the wind, you can determine the resultant velocity of the plane in each scenario.
1. X = 10 mi/h.
Y = 80 mi/h.
Vr = Sqrt(X^2 + Y^2).
2. Same Procedure as #1.
3. Vr = Vp + Vw = 80 + 10 = 90 mi/h.
4. Vr = Vp + Vw = 80 + (-10) = 70 mi/h.