Flight Patterns

By: Saumya Naik

The Situation

A hummingbird is sitting on a branch that is 16m above ground. The hummingbird sees its flock leaving, so it flies away to join them. The rest of the flock is 60m above ground. A hawk sees its prey on the ground and is descending to catch it, from a height of 100m. Which bird will reach it's goal first?

Variables

Variable X - Time (seconds)

Variable Y - Height from ground (meters)

Hawk

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Hummingbird

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Equation for Hummingbird

Find Slope with two points given. (0,16) (2,20)


  • Y2- Y2 / X2-X1= slope
  • 20-16/2-0= slope (sub in the given points)
  • 4/2= slope
  • 2=slope

Now find the value of b using y=mx+b

  • y= mx+b
  • y=2x+b
  • 16=2(0) + b (sub in given point (0,16) to find the value of b)
  • 16=b



Therefore the equation is y= 2x+16

Equation for Hawk

Find slope with two points given. (0,100) (2, 90)


  • Y2-Y1 / X2-X1 = slope
  • 90-100 / 2-0 = slope (sub in the given points)
  • -10 / 2 = slope
  • -5 = slope

Find the value of b using y= mx+b


  • y= mx+b
  • y= -5x+b
  • 100= -5(0) +b (sub in given point (0,100) to find the value of b)
  • 100=b


Therefore the equation is y= -5x + 100

Graph

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Close up of Intersection

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Hawk in red

Hummingbird in blue


Method 1: Graphing

After graphing both lines you can see that both birds will meet at 12 seconds and both birds will be 40m above ground


Method 2 substitution:

-5x+100= 2x+16

100-16=2x+5x

84=7x

84/7= x

12=x

Now sub one given point.

y= 2(12) +16

y= 24+16

y= 40


So the point is (12,40)

Meaning that they will meet at 12 seconds at the height of 40m above ground

Summary

Going back to the question in the beginning:

Which bird will reach it's goal first?

Looking at the graph you can tell that the hawk will catch his prey before the hummingbird catches up with its flock. The hawk will reach it's prey 2 seconds earlier than the bird reaches it's goal. The hawk flies faster than the hummingbird.