A supersonic jet travels at 3.0 Mach (i.e. at three times the speed of sound). It cruises at 20,000 m above the ground. we choose t=0 when the plane passes directly overhead of an observer. At what time t will the observer hear the plane?

-- there is a diagram of a person standing on the ground, a plane above him and the distance from him to the plane(vertial) is h (20,000 m). there is a line from the tail of the plane( x axis direction) and and a line from the plane to the ground, at an angle theta to the x axis..

.. i am extremely confused with this question, any guidance would be helpful.

The sound reaches you at the speed of sound after the plane passes overhead. You will not hear the plane until it has gone much further since it is going faster than a sound wave but it still takes the sound 20,000 meters/speed of sound to reach you from overhead.

The triangle shows where behind the plane the plane can be heard.
You will hear a loud bang because all the sound that would normally be radiated in front of the plane is concentrated along that line. It can not get ahead of the line because the line is moving faster than sound.

To determine at what time the observer on the ground will hear the supersonic jet, we need to consider the speed of sound and the distance between the observer and the plane.

Let's break down the problem into two steps:

Step 1: Find the distance between the observer and the plane at time t.
At any given time t after passing directly overhead, the plane will have traveled a certain distance. We can represent this distance as x(t), which is a function of time. To find x(t), we need to consider the speed of the plane. Given that the jet travels at 3.0 Mach, we know its speed is three times the speed of sound. We can use the formula:
x(t) = (speed of the jet) * t

However, we need to consider that the plane is not moving directly away from the observer but at an angle, which is represented as theta in the diagram you provided. To account for this, we multiply the speed of the jet by the cosine of theta. So the updated formula becomes:
x(t) = (speed of the jet) * cos(theta) * t

Step 2: Calculate the time it takes for the sound to travel the distance between the observer and the plane.
The speed of sound is approximately 343 m/s (at the average temperature of 20 degrees Celsius). Since the sound waves need to cover the distance between the observer and the plane (h) to reach their ears, we can use the formula:
t_sound = h / (speed of sound)

Now that we have both the time it takes for the plane to travel the distance (t) and the time it takes for the sound to travel the distance (t_sound), we can set them equal to each other and solve for t:
(speed of the jet) * cos(theta) * t = h / (speed of sound)
t = (h / (speed of sound)) / ((speed of the jet) * cos(theta))

Using the given values (h = 20,000 m, speed of sound = 343 m/s, speed of the jet = 3 * speed of sound), and knowing the value of theta (coming from the diagram), you can substitute these values into the equation to find the time t.