A rocket is launched vertically from ground with a constant acceleration. If the rocket emits a burst of sound every five seconds after launching, find the difference of intensity levels observed at the launching site for the first and the second bursts of sound.
To find the difference of intensity levels observed at the launching site for the first and second bursts of sound, we need to understand how sound intensity changes with distance and time.
The intensity of sound is given by the formula:
I = P / A
where I is the intensity, P is the power, and A is the area over which the sound is spread.
In this case, we are given that the rocket emits a burst of sound every five seconds after launching. This means that the time between the first and second bursts of sound is also five seconds.
Now, let's look at how the distance between the rocket and the launching site changes over time. Since the rocket is launched vertically with a constant acceleration, we can use the equations of motion to describe its position.
The equation for the position of the rocket at any time t is given by:
s = ut + (1/2)at^2
where s is the position, u is the initial velocity, a is the acceleration, and t is the time.
Since the rocket is launched vertically and we are interested in the difference of intensity levels at the launching site, we can assume that the rocket starts from the ground, so the initial position (s₀) is zero.
Therefore, the position of the rocket at time t is:
s = (1/2)at^2
Now, let's calculate the position of the rocket at the time of the first and second bursts of sound.
For the first burst of sound, t = 5 seconds. Substituting this value into the equation for position, we get:
s₁ = (1/2)a(5^2) = 12.5a
For the second burst of sound, t = 10 seconds. Substituting this value into the equation for position, we get:
s₂ = (1/2)a(10^2) = 50a
Now, let's calculate the difference in intensity levels. Remember that intensity is inversely proportional to the square of the distance.
The difference in intensity levels is given by the formula:
ΔI = (I₁ / r₁²) - (I₂ / r₂²)
where I₁ and I₂ are the intensities at distances r₁ and r₂, respectively.
Since the rocket is launched vertically from the ground, the distance of the rocket from the launching site at the time of the first and second bursts of sound would be equal to the position of the rocket at those times. So, r₁ = s₁ and r₂ = s₂.
Plugging in the values we calculated earlier, the difference in intensity levels is:
ΔI = (I₁ / (12.5a)²) - (I₂ / (50a)²)
Now, we need additional information about the power emitted by the rocket during the bursts of sound (P) and the area over which the sound is spread (A) to calculate the intensity levels and the difference between them.