A plane flies at 1.25 times the speed. It's sonic boom reaches aman on the ground 1.00 min after the plan passes directly overhead.What is the altitude of the plane? Assume the speed of sound to be 330m/s.
The Mach number is 1.25. (You left out the words "of sound" in your first sentence). The Mach number can be used to get the approximate angle the shock wave makes with the horizontal. (It also depends somewhat upon the "wedge angle" of the airplane). The APPROXIMATE shock wave angle is
theta = sin^-1 (1/M) = 53.1 degrees. When the shock wave is felt, the plane has travelled 1.25*330*60s past the overhead point, or 4.95 km.
Draw yourself a figure of the situation.
The altitude H is given by
H/4.95 km = tan 53.1 = 1.332
H = 6.59 km
This problem is not as simple as they would have you believe. The shock wave angle formula is an approximation.
To find the altitude of the plane, we need to understand the relationship between speed, time, and distance.
Let's denote the speed of the plane as v and the altitude as h. We know that the sonic boom takes 1.00 min (or 60 seconds) to reach the man on the ground.
The speed of sound is given as 330 m/s. Therefore, in 1.00 min (60 seconds), the sonic boom will travel a distance of: distance = speed × time = 330 m/s × 60 s = 19,800 meters.
Now, let's focus on the plane. Since the plane flies at 1.25 times the speed of sound, its speed can be calculated as follows: v = 1.25 × 330 m/s = 412.5 m/s
Since the sonic boom reaches the man on the ground 1.00 min after the plane passes directly overhead, we can calculate the distance the plane has traveled during that time. This distance is equal to the speed of the plane multiplied by the time it took to reach the man: distance = speed × time = 412.5 m/s × 60 s = 24,750 meters.
Now, let's visualize the scenario. When the sonic boom reaches the man on the ground, it has traveled a distance of 19,800 meters. During this time, the plane has moved ahead and traveled a distance of 24,750 meters. Therefore, the total distance between the plane and the man on the ground is the sum of these distances: distance = 19,800 m + 24,750 m = 44,550 meters.
However, this distance is not equal to the altitude of the plane. We need to consider the right-angled triangle formed by the altitude (h) and the ground distances traveled by the sonic boom and the plane.
Using the Pythagorean theorem, we can find the altitude of the plane:
h^2 = distance^2 - (distance traveled by sonic boom)^2
h^2 = 44,550 m^2 - 19,800 m^2
h^2 = 1,976,002,500 m^2
h = sqrt(1,976,002,500) m
h ≈ 44,441 meters
Therefore, the altitude of the plane is approximately 44,441 meters.