Relative Motion in vectors form

In summary, the bird must fly at a bearing of 52 degrees relative to the air in order to reach its nest, taking into account the wind blowing from the northwest at 70 km/h and the bird's capability of flying at 80 km/h. This can be solved by using the relationship \vec{V_{B/G}} = \vec{V_{B/A}} + \vec{V_{A/G}} and finding the components of \vec{V_{B/A}} using the given information.
  • #1
garytse86
311
0
The question is as follow:
A bird is capable of flying at 80km/h. It wishes to fly to its nest which is due East of its present position. Thre is a win blowing from the northwest at 70km/h. Find the direction relative to the air in which the bird must fly to reach its nest.

I tackled the problem like this:
1) Velocity of air = (35root(2))i + (-35root(2))j
Resultant velocity (=velocity of bird) = due east
Velocity of bird relative to air = 80j

Is this correct?

actually I think the method is correct as well, but surely the bird cannot fly just in the north direction (like a helicopter).

Or should I think like this:
2) for i direction: 35root(2) = 80cos(theta) - this gives the right angle tho.
or:
3) for j direction: 35root(2) = 80sin(theta) - I believe this method is correct because you want the resultant direction due east, therefore no j component of the resultant velocity.

Please help.
Gary
 
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  • #2
Let B=bird, A=air, G=ground, and B/A = "bird with respect to ground", etc. The relationship among the relative velocities is: [itex]\vec{V_{B/G}} = \vec{V_{B/A}} + \vec{V_{A/G}}[/itex]. [itex]\vec{V_{A/G}}[/itex] is given, and you know that [itex]\vec{V_{B/G}}[/itex] must have a direction of east. Solve for the components of [itex]\vec{V_{B/A}}[/itex], knowing that it has a magnitude of 80 km/h.
 
  • #3
I did: v(b/a) = v(b) - v(a)
v(b/a) = 80i - ((30root2)i - (30root2)j)
v(b/a) = (80-30root2)i = (30root2)j
and got an angle of 48.5 degrees. Is this correct? (but the book says it should be 52 degrees)
 
  • #4
garytse86 said:
The question is as follow:
A bird is capable of flying at 80km/h. It wishes to fly to its nest which is due East of its present position. Thre is a win blowing from the northwest at 70km/h. Find the direction relative to the air in which the bird must fly to reach its nest.

I tackled the problem like this:
1) Velocity of air = (35root(2))i + (-35root(2))j
Resultant velocity (=velocity of bird) = due east
Velocity of bird relative to air = 80j

Is this correct?

actually I think the method is correct as well, but surely the bird cannot fly just in the north direction (like a helicopter).

Or should I think like this:
2) for i direction: 35root(2) = 80cos(theta) - this gives the right angle tho.
or:
3) for j direction: 35root(2) = 80sin(theta) - I believe this method is correct because you want the resultant direction due east, therefore no j component of the resultant velocity.

Please help.
Gary

The fact that the bird "is capable of flying at 80km/h" tells you that the speed of the bird (the length of the velocity vector) is 80. It does NOT tell you the direction: that's the whole point of the problem. Draw a picture: the bird wants to fly due east so draw a line in that direction. The wind is 70 km/h from the nw so draw a line like that ending at the tip of the east line. Now draw a line from the beginning of the east line to the beginning of the "nw" line. That's the triangle you need to solve. You know the lengths of two sides (80 and 70) and you can know one angle (45 degrees) so you can solve for the other angles.

By the way, I'm very puzzled by your remark "surely the bird cannot fly just in the north direction (like a helicopter)."

? are you sure you know which direction north is?! Birds fly north every spring!
 
  • #5
i tried that but didnt get the right answer. I was being stupid for the north direction thing - thought in two dimensions there are only up and down, and from west to east.
 
  • #6
garytse86 said:
I did: v(b/a) = v(b) - v(a)
v(b/a) = 80i - ((30root2)i - (30root2)j)
v(b/a) = (80-30root2)i = (30root2)j
and got an angle of 48.5 degrees. Is this correct? (but the book says it should be 52 degrees)
Your error is taking the speed of the bird with respect to the ground to be 80. True, it does travel east. But 80 km/h is its speed with respect to the air, not the ground.

Hint: find the y-component of the bird's velocity with respect to the air.
 
  • #7
i got:
v(a) + v(b/a) = v(b)
(35root2)i - (35root2)j + xi + yj = zi
but v(b) has no j component: therefore y = 35root2
x(square) + 35root2(square) = 80(square)
x = root2950
tan(theta) = 0.78756153 --> theta = 38.2 degrees
so the bearing is 52 degrees as required.
 
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1. What is relative motion in vectors form?

Relative motion in vectors form is a way to describe the motion of an object in relation to another object, using mathematical vectors. It takes into account the position, direction, and speed of both objects to determine their relative motion.

2. How is relative motion in vectors form different from other ways of describing motion?

Relative motion in vectors form is different from other ways of describing motion because it considers the motion of two objects in relation to each other, rather than just the motion of one object. It also uses mathematical vectors, which represent both the magnitude and direction of motion.

3. What are some real-world applications of relative motion in vectors form?

Relative motion in vectors form has many practical applications, such as in navigation and transportation, where the motion of a moving object must be calculated in relation to other objects or landmarks. It is also used in physics and engineering to analyze the motion of objects in complex systems.

4. How do you calculate relative motion in vectors form?

To calculate relative motion in vectors form, you need to determine the position, direction, and speed of both objects. Then, you can use vector addition and subtraction to find the resultant vector, which represents the relative motion between the two objects.

5. Are there any limitations to using relative motion in vectors form?

While relative motion in vectors form is a useful tool for analyzing motion, it does have some limitations. It assumes that the motion of both objects is linear and that there are no external forces acting on them. In reality, the motion of objects is often more complex, and there may be other factors that affect their motion.

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