Supersonic plane at mach 3

[ScienceGeek24]

Homework Statement

A supersonic plane flies at Mach 3 at an altitude of 20,000 m. A person on the ground sees the plane directly overhead. How much time passes the before she hears the sonic boom?

Homework Equations

f=fo(v+-vo/v-+vs)

The Attempt at a Solution

I don't really know how to start up with this one, I was hoping for some help here.

 

[PeterO]

Homework Statement

A supersonic plane flies at Mach 3 at an altitude of 20,000 m. A person on the ground sees the plane directly overhead. How much time passes the before she hears the sonic boom?

Homework Equations

f=fo(v+-vo/v-+vs)

The Attempt at a Solution

I don't really know how to start up with this one, I was hoping for some help here.

The sonic boom is just a noise produced by the plane, which has to travel 20,000m to reach you.
The answer would be the same if it was a helicopter hovering at 20000m then firing a really loud gun - or with a really big sound system playing AC-DC.

 

[ScienceGeek24]

So what equation do you suggest I can use to start up the problem?

 

[PeterO]

So what equation do you suggest I can use to start up the problem?

You want to calculate the time taken for sound to travel 20000m. What equation do you think you need - it connects speed, time and distance?

You will need to know the speed of sound - and will probably have to make the assumption that it remains constant all the way from up there to down where you are!

 

[asteriatic]

I would approach this problem by first understanding the concept of Mach number and how it relates to the speed of sound. Mach number is a dimensionless quantity that compares the speed of an object to the speed of sound in the surrounding medium. At Mach 1, the object is traveling at the speed of sound, and at Mach 3, the object is traveling three times the speed of sound.

In this problem, we are given that the plane is flying at Mach 3 at an altitude of 20,000 m. This means that the plane is traveling at three times the speed of sound, which is approximately 343 m/s at sea level. However, the speed of sound decreases with altitude, so at an altitude of 20,000 m, the speed of sound would be lower than 343 m/s.

To calculate the time it takes for the sonic boom to reach the person on the ground, we can use the formula provided in the homework section. This formula calculates the frequency of the sound wave (f) based on the observer's frame of reference and the source's frame of reference. In this case, the observer is the person on the ground and the source is the supersonic plane.

Plugging in the values, we get:

f = fo (v + vs) / (v - vo)
Where:
f = frequency of the sound wave
fo = original frequency of the sound wave
v = speed of sound at the plane's altitude
vs = speed of the plane
vo = speed of sound at the observer's location (ground level)

To solve for time, we can use the formula t = 1/f, where t is time and f is frequency. So, the time it takes for the sonic boom to reach the person on the ground would be:

t = 1 / [fo(v + vs) / (v - vo)]

To solve for the frequency (fo), we can use the formula fo = vs / λ, where λ is the wavelength of the sound wave. The wavelength can be calculated using the formula λ = v / f, where v is the speed of sound and f is the frequency. Substituting this into the original formula, we get:

t = 1 / [(vs / λ)(v + vs) / (v - vo)]

Now, we can plug in the values and solve for time. However, it is important to note that this calculation will only give us an