Thread: Trigonometric Applications - Help With 2 Word Problems

Hello,

I have some questions regarding a couple of questions on an important homework assignment, and because my instructor hastily finished her lecture on vectors at the end of class, I'm really confused on two word problems (applications).

If someone could help me visualize/setup the problem, and guide me with solving it I would greatly appreciate it. Particuarily the 2nd one I am having the most difficulty conceptualizing.

Problem #1:
"A plane has an airspeed of 245 mph and a bearing of N 40 degrees E. The air currents are moving at a constant speed of 32.5 mph in the direction of S 50 degrees E. Find the ground speed and the actual course of the plane."

Problem #2:

"John and his sister Kathy have attached a rope to the branch of a tree and tied a board to the other to form a swing. John sits on the board while Kathy pushes him through an angle of 23.4 degrees and holds him there. If John weighs 97.5 pounds, find the magnitude of the force Kathy must push him horizontally to keep John in static equilibrium."

Let me know if clarification is needed on any of these.

Thanks in advance for the help!

Originally Posted by Do.The.Math

If someone could help me visualize/setup the problem, and guide me with solving it I would greatly appreciate it. Particuarily the 2nd one I am having the most difficulty conceptualizing.

Problem #2:

"John and his sister Kathy have attached a rope to the branch of a tree and tied a board to the other to form a swing. John sits on the board while Kathy pushes him through an angle of 23.4 degrees and holds him there. If John weighs 97.5 pounds, find the magnitude of the force Kathy must push him horizontally to keep John in static equilibrium."

First, assume we're dealing with an 'ideal' situation in which all other forces are negligible (like the weights of the rope and seat). Second, can we assume Kathy is pushing perpendicular to the 24.3 degree angle? That is, is he sitting straight and is she pushing direclty against his back? If so, the free body diagram for this problem is fairly straightforward as shown in the attachment.

F (the force with which Kathy is pushing) must be large enough to cancel the vertical force W (the weight of John).

You can use right-triangle trig to figure out then numbers, including the answer to your question. This solution will give you the horizontal component of the force necessary to keep him in position.

Originally Posted by Do.The.Math

Problem #1:
"A plane has an airspeed of 245 mph and a bearing of N 40 degrees E. The air currents are moving at a constant speed of 32.5 mph in the direction of S 50 degrees E. Find the ground speed and the actual course of the plane."

The plane is travelling on a certain bearing but is also being pushed around by the wind. The overall (ground) speed is the sum of the vectors describing these.

When adding vectors, always position them head-to-tail in your drawings. And for this problem you'ld better know what angles are being given. In any problems dealing with directional bearings, you can put a N/S/E/W coordinate system wherever you want.

Take a look at the attached drawing (p is the angle of the plane and P is it's airspeed, while w is the angle of the wind and W is it's speed). Can you use the given angles to find a right triangle you can work with to answer the questions?

Thanks PFLO!

I was able to solve the problems with your help in visualizing the vectors for me. I have provided the solutions in case anyone was interested.

#1.) Find the magnitude of the planes' actual ground speed and the true course (direction/bearing) of the plane. I.E. the vector of the plane (magnitude & direction).

|V| = Sqrt ( |Vx|^2 + |Vy|^2 )
|V| = Sqrt ( 32.5^2 + 285^2 )
|V| = Sqrt (82281.25)
|V| ~ 287 miles per hour = ground speed.

Bearing = N 40 degrees + Theta Angle E

Theta = tan^-1 ( |Vx| / |Vy| )
Theta = tan^-1 ( 32.5 / 285 )
Theta ~ 6.5 degrees

True vector of the plane: 287 miles per hour traveling in the direction N 46.5 degrees E.


#2.) Find the horizontal force Kathy must extert to keep John in static equilibrium (in the same place).

tan 23.4 degrees = |F| / |W|
tan 23.4 degrees = |F| / 97.5 lbs

|F| = 97.5 * tan 23.4 degrees
|F| ~ 42.2 lbs.

Hence, Kathy must exert 42.2 lbs of horizonal force to keep John in equilibrium.

Note: the horizontal keyword is important, as this creates a right triangle when the vectors are drawn out (contrary to the vector drawn in the thumbnail provided by PFLO; if it was in the case of PFLOs' vectors, solving this problem would require the use of the law of sines or cosines).

If any of this information/solving is incorrect, please let me know! You won't be able to understand the solution without first drawing the vectors (or at least understanding them) & reading the complete word problems in the 1st post.

Originally Posted by Do.The.Math

Note: the horizontal keyword is important, as this creates a right triangle when the vectors are drawn out (contrary to the vector drawn in the thumbnail provided by PFLO; if it was in the case of PFLOs' vectors, solving this problem would require the use of the law of sines or cosines).

If any of this information/solving is incorrect, please let me know!

It actually wouldn't require the use of the law of sines or cosines. The triangle you would use to solve it is a right triangle, with F being the hypotenuse and W being the side opposite the 23.4-degree angle. Therefore all you need is right-triangle trig (SOH, CAH, TOA). This is why I included the assumption that Kathy's force is perpendicular to John's back.

But your assumption, that Kathy's force is directly entirely horizontally, is just a valid.