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3.1) Busy Bees

A worker bee is ready to fly out and gather pollen, but on her way out of the hive she meets one of her fellow workers who is just returning. The incoming bee tells her, in bee-dance language, that there is a good pollen source 0.3 km southeast of the hive. However, before the outgoing bee leaves the hive, she meets another incoming bee and she tells her that there is also a pollen source 0.5 km due south of the hive. If the busy worker is to visit both of these pollen sources in their respective order before returning to the hive, what direction and distance will she need to fly to get from the first source to the second source?  If her average velocity is about 1.5 m/s, how long will it take her to get from the first to the second source?

A flock of songbirds in the western U.S. is preparing to migrate south for the winter. The winds at their migration altitude are from the southwest—235o  by the compass, with wind velocities averaging at 10 kts. What direction should they actually fly in order to end up due south in Mexico? If their in-flight average velocity is 17.4 kts, how many days will it take them to travel the distance to their winter habitat there, about two thousand miles away? Assume that they can travel 17 hours out of each day.  (1 kt = 1 nautical mile per hour; 1 nautical mile = 1.15 statute miles)

3.3) Mrs. Gibson

You are one of a team of researchers interested in flight patterns and speeds of the wandering albatross. Using tracking devices you are able to determine that “Mrs Gibson” (a female wandering albatross of the Aukland island plumage—Diomedea exulans gibsoni—used in this particular study) is moving at 46km/h towards the east-northeast from New South Wales over the Tasman Sea with a compass heading of 77o.  From atmospheric data you know that the winds are coming at her from just north of west at 275o at 32 km/h. How much is the wind contributing to the eastward component of her velocity? How much is the wind detracting from her northward velocity? At what heading and with what airspeed is the albatross flying?

3.4) West with the Night

It is September of 1936, and you are the Kenyan aviatrix Beryl Markham; today you are attempting the first solo flight of the Atlantic from east to west in a specially-outfitted Vega Gull, against the prevailing winds of the North Atlantic jet stream. You don't have a radio to call for help, so you'd better do some calculations or you might not end up over land when your specialized fuel tanks run out! Your plan is to fly from London to New York, proving the potential for commercial airline routes. The initial  heading from London to New York is 288 degrees. At 10,000 ft, the winds are coming at you from the southwest at 235 degrees with wind velocity of about 50 kts. If your valiant blue and silver Gull flies at its top airspeed of 163 mph, at what heading should you initially fly to correct for the wind?  What will be your speed relative to the ground?

3.5) Las Hermanas de las Montañas

Katy-Robin Garton

Katy and her sister Rachel are climbing Mt. Denali in Alaska.  Every evening they plot their position on a topo map, calculate the horizontal distance they have traveled, and record this and their elevation.  This chart is shown below.

 Day Location Elevation (ft) Horizontal Distance (miles) 1 Base Camp 7200 0.0 2 Camp 1 7800 5.5 3 Camp 2 9700 2.5 4 Camp 3 11,000 2.25 5 Camp 4 14,280 2.75 6 Ridge Camp 16,200 2.0 7 High Camp 17,200 0.75 8 Summit 20,328 2.5

a)      On what day did they travel the farthest?  (Total horizontal plus vertical?)

b)      On what day did they gain the most elevation?

c)      What is the total (horizontal plus vertical) distance they traveled?

d)      What is their average distance per day?

e)      What is their average elevation gain per day?

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3.6) Treeleaf Vector Lab

After it rains, the water that is absorbed by the ground and subsequently by a tree’s root system must travel through the tree’s trunk and branches to get to the leaves.  Find a tree with reasonably straight branches, and find at least one leaf on this tree whose water-deliverance path you can figure out (and reach). Take a protractor and a tape measure and measure the distances and angles of each branch connecting the base of the trunk to this leaf.

What is the total upward displacement of the water that travels from just below ground surface to this leaf? What is the entire displacement? If the water in the xylem of a tree travels at 1 mm/s, how long will it take to make this journey?

3.7) 3-D Gnomon Lab

We can use vectors to know more about our motion relative to the sun.  How does the sun move across the sky?  Where does it rise? Set?  How do we explain its path?  How does the path of the sun change in the sky as we move to different spots on the Earth?  One way that we can answer these types of questions is by observing the shadows of objects of known heights over a period of a day.  One way that this can be done is by constructing a Gnomon and observing its shadow.  A Gnomon is a pillar constructed perpendicular to level ground (and the horizon).  By recording the tip of the shadow of the gnomon at different times during the day, we can answer the questions that were initially posed.

The short and sweet version of this includes constructing a Gnomon on a flat surface, drawing an accurate circle around the gnomon, with the gnomon at the center of that circle.  From 10AM until 2Pm record the location of the shadow and the time of that shadow.  Note specifically where the shadow enters and exits the circle and record those times.  Devise some coordinate system to record the positions of the each shadow so that you can carry your data around in a notebook instead of risking the loss of your data to a small squirrel or passing bicyclist.

Now with your time and location data determine a few things about the relative motion of the Earth and the sun.

a.       Where is the sun in relation to earth?  Plot the position of the shadow as a function of time.   Where is East? West? North? South? Use the circle that you drew around the Gnomon to figure this out.  Show and describe your methods.

b.      What is your latitude if you know the relative angle of the sun? (this is easiest around the solstice or the equinox, but data could be found for other times as well)  Conversely, if you know the latitude, at what angle is the Earth tilted relative to the sun?  Compare these findings to the angle that we make with Polaris.

c.       Does the sun move across the sky in a plane?  Determine this by constructing vectors which point towards the sun  and then taking their cross-products to determine if they are in a plne.

d.      Is the velocity of the sun as it moves across the sky constant?  Use again the vectors that point towards the sun and then subtract them to get velocity vectors.  Are these velocity vectors the same size?

e.       Can we figure out how far away the sun is?  Why or why not?

Where does the sun rise and set?  Due East-West? How would that change if we were   on the equator?

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Kinematics Problem Sets

Posted on 8/1/05