Sound of overhead plane is heard in the distance

[jamiewilliams]

Homework Statement

A jet is flying horizontally [drawing shows an observer on the ground, a plane overhead at point B, and an outline of the plane from it's observed position to the left at point A. The angle between the observer and point A is 36°.] When the plane is directly overhead at B, a person on the ground hears the sound coming from A in the drawing. The average temperature of the air is 20 degrees Celsius. If the speed of the plane at A is 166 m/s, what is its speed at B, assuming that it has a constant acceleration?

Homework Equations

? No Clue

The Attempt at a Solution

I am not sure which equation(s) to use. I know that the temperature is not relevant to the equation, that it gives the velocity of sound at 343 m/s. Other than that, I am at a total loss for where to start on this. Please help!

 

[cepheid]

The KEY piece of information imparted by this problem is that:

the time taken for the sound to travel from the point A to the ground is the same as the time taken for the plane to travel from point A to point B.

You can figure out the first thing using the speed of sound, and the distance traveled by the sound.

You can then use that time to figure out the second part of the problem, which is a simple kinematics problem:

a plane travels from point A to point B in time t (which we solved for above by computing the travel time of the sound wave). If it moves this distance at acceleration a, and its initial velocity is 166 m/s, how fast is it going at the end of the motion?

You have two unknowns here: the acceleration and the final speed. But it's okay, because you also have two equations: the equation for speed vs time (at constant acceleration), and the equation for distance vs. time (at constant acceleration)

All of the above assumes that the distance between A and B are given, as is the height of point B. If they aren't, I don't know how you'd solve it (you wouldn't be able to get a numerical answer, anyway. You could express the answer algebraically in terms of the distance between A and B).

 

[jamiewilliams]

All of the above assumes that the distance between A and B are given, as is the height of point B. If they aren't, I don't know how you'd solve it (you wouldn't be able to get a numerical answer, anyway. You could express the answer algebraically in terms of the distance between A and B).

No distances are given. The only given data is velocity at point A, θ=36.0, and the speed of sound. I know the answer is 237 m/s, but I have no idea how to get there.

 

[cepheid]

No distances are given. The only given data is velocity at point A, θ=36.0, and the speed of sound. I know the answer is 237 m/s, but I have no idea how to get there.

Okay, you can solve it, but the algebra is just a bit more long and tedious than I was expecting. It's no problem that the distance from A to B is not given. Call it "x."

Now call the distance from point A to the observer "d." This is the distance traveled by the sound wave. The distance d is related to x by simple trigonometry. So basically, x is the only unknown so far.

The travel time t = d/(sound speed). So, basically, t also depends only on x. So the only unknown so far is still just x.

Now you have the kinematics equations I mentioned before. The easiest ones to use are the one that expresses the final speed in terms of initial speed, acceleration, and time (or x).

The second equation that is handy to use relates the square of the final speed to the square of the initial speed, the acceleration, and the distance x travelled.

Using these two equations, you should be able to solve for the final speed. Again, the algebra is a bit of a pain, but it is just math.

EDIT: I should mention that you'll get a quadratic, but you can just discard one of the solutions as unphysical.

EDIT 2: I had to assume that the 36 degrees was measured from the vertical, rather than from the horizontal, in order to get the right answer. Hopefully that is how it is in the original diagram.

 

[jamiewilliams]

I had to assume that the 36 degrees was measured from the vertical, rather than from the horizontal, in order to get the right answer. Hopefully that is how it is in the original diagram.

it is :)

 

[jamiewilliams]

I feel like this may be a silly question, but I am not sure what to use in the kinematic equations for acceleration? I know that acceleration is constant but I don't know what to use for its value.

 

[cepheid]

I feel like this may be a silly question, but I am not sure what to use in the kinematic equations for acceleration? I know that acceleration is constant but I don't know what to use for its value.

You'll find that you won't need it in the end. Don't plug in numbers from the start. Instead, manipulate the equations algebraically to solve for v_final. This involves using each equation successively to eliminate an unknown until in the end you're only left with the unknown that you're trying to find: v_final. Once you have an expression for v_final, then you can plug in the numbers in order to compute it.