Ground speed of a plane - vectors

In summary, the conversation revolved around finding the groundspeed of a plane heading due east while climbing at a rate of 80kph, with a speed of 480kph and a wind blowing at a speed of 100kph to the northeast. The two attempted solutions involved calculating the velocity relative to the air and then adding the wind velocity to find the groundspeed. The correct answer was found to be approximately 549kph, with a small discrepancy due to early rounding. Precision was emphasized as important in finding accurate results.
  • #1
tinkus
13
0

Homework Statement



A plane is heading due east and climbing at he rate of 80kph. if its speed is 480kph and there's a wind blowing 100kph to the northeast, wha is the groundspeed of the plane

Homework Equations





The Attempt at a Solution


w=100cos45+100sin45= 70i+70j
480= vxi+ 70j+80k
vxi= 468
Groundspeed= 468i+70i+70j=538i + 70j = 543kph

or


480= vxi+70j
vxi= 475
Groundspeed = 475i +70i+70j = 545i + 70j = 549kph
 
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  • #2


tinkus said:

Homework Statement



A plane is heading due east and climbing at he rate of 80kph. if its speed is 480kph and there's a wind blowing 100kph to the northeast, wha is the groundspeed of the plane

Homework Equations





The Attempt at a Solution


w=100cos45+100sin45= 70i+70j
480= vxi+ 70j+80kk
How do you get the "70j" here?

vxi= 468
Groundspeed= 468i+70i+70j=538i + 70j = 543kph

or


480= vxi+70j
vxi= 475
Groundspeed = 475i +70i+70j = 545i + 70j = 549kph

Again, where did you get the "70 j" as part of the airplane's velocity? If the airplane is going "due east" shouldn't it be 0j?
 
  • #3


yea i thought as much but isn't the y-component of the windspeed part of the airspeed? Can you show me how to solve this problem?
 
  • #4


No, the "airspeed" is the speed through the air and is separate from the wind speed. And you surely can't have thought you should include the j (north-south) component but not the i (north-south) component?

The velocity relative to the air is vx i+ 80 k and the airspeed is 480 so [itex]vx^2+ 80^2= 480^2[/itex]. Once you have found that, the velocity relative to the ground is (vx+ 70)i+ 70j (the k component is not relevant to moving relative to the ground).
 
  • #5


ok thanks, i got 547.8kph. the answer key is 548.6, i guess is still correct...
 
  • #6


You lost some precision by the very rough rounding you did early on.
100 [itex]\sqrt{2}[/itex]/2 is closer to 71 than it is to 70.

You will always get more precise results if you refrain from rounding until your final result.
 
  • #7


Thanks, it was a complete oversight...duly noted
 

FAQ: Ground speed of a plane - vectors

What is the ground speed of a plane and how is it different from airspeed?

The ground speed of a plane refers to the speed at which the plane is moving over the ground, while airspeed refers to the speed at which the plane is moving through the air. Ground speed takes into account the effects of wind on the plane's movement, while airspeed does not.

How is the ground speed of a plane calculated?

The ground speed of a plane is calculated by combining the airspeed of the plane with the speed and direction of the wind. This is done using vector addition, which takes into account the magnitude and direction of both vectors to determine the resulting ground speed.

Why is it important for pilots to know the ground speed of their plane?

Knowing the ground speed of a plane is important for pilots to accurately plan their flight time and fuel consumption. It also helps them to adjust their airspeed and navigation to account for wind conditions and ensure a safe and efficient flight.

How do variations in wind speed and direction affect the ground speed of a plane?

Wind speed and direction can have a significant impact on the ground speed of a plane. If the wind is blowing in the same direction as the plane's movement, it will increase the ground speed. Conversely, if the wind is blowing in the opposite direction, it will decrease the ground speed. Changes in wind direction can also affect the direction of the ground speed vector.

Can the ground speed of a plane ever be greater than its airspeed?

Yes, the ground speed of a plane can be greater than its airspeed under certain wind conditions. If the wind is blowing in the same direction as the plane's movement, it will add to the airspeed and increase the ground speed. This is known as a tailwind and can result in a higher ground speed. However, it is important for pilots to be aware of potential hazards such as turbulence and crosswinds that can also be caused by strong winds.

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