Suppose we launch a scientific rocket straight up from sea level (at standard conditions) with a
pressure measuring device attached to it. Given that in the first 12 seconds the rocket experiences
a 25% decrease in air pressure, what is the average speed (rate of climb) of the rocket in these
first 12 seconds?
Well, that rocket must be quite the professional vacuum cleaner if it can decrease air pressure by 25% in just 12 seconds! But hey, who needs regular pressure when you can have a space vacuum, am I right? In any case, to calculate the average speed or rate of climb of the rocket, we would need more information like the starting pressure and the final pressure. So, until we get those details, let's just say the rocket is defying gravity with its swift ascent into the realm of low pressure!
To find the average speed (rate of climb) of the rocket in the first 12 seconds, we can use the formula:
Average speed = Total distance traveled / Total time taken
In this case, we need to calculate the total distance traveled by the rocket. Since the rocket is launched straight up, the total distance traveled is equal to the altitude gained.
To calculate the altitude gained, we need to know the initial and final air pressures. Let's assume that the initial air pressure is P1 and the final air pressure is P2.
Given that the rocket experiences a 25% decrease in air pressure, the final air pressure can be calculated as:
P2 = P1 - (25/100) * P1
= P1 - 0.25 * P1
= 0.75 * P1
Since air pressure decreases with altitude, we can assume that the initial air pressure (P1) is the air pressure at sea level, known as standard atmospheric pressure. At sea level, the standard atmospheric pressure is approximately 1013.25 hPa.
P1 = 1013.25 hPa
Next, we can convert the pressure from hPa to Pascals for consistency in units.
P1 = 1013.25 hPa * 100 Pa/hPa
= 101,325 Pa
Now, we can calculate the altitude gained (h) using the relationship between pressure and altitude:
h = (P1 - P2) / ρ * g
Where:
- ρ is the density of air
- g is the acceleration due to gravity
The density of air (ρ) varies with altitude but for simplicity, we can assume a constant value of ρ = 1.225 kg/m³ at sea level.
Using a standard value for the acceleration due to gravity (g) of 9.8 m/s², the equation becomes:
h = (101,325 Pa - 0.75 * 101,325 Pa) / (1.225 kg/m³ * 9.8 m/s²)
= 25281.25 Pa / (11.98 kg/(m*s²))
= 2114.24 m
Therefore, the rocket gains an altitude of approximately 2114.24 meters in the first 12 seconds.
Finally, we can calculate the average speed (rate of climb) of the rocket using the equation:
Average speed = Total distance traveled / Total time taken
In this case, the total distance traveled is 2114.24 meters, and the total time taken is 12 seconds.
Average speed = 2114.24 m / 12 s
≈ 176.186 m/s
Therefore, the average speed (rate of climb) of the rocket in the first 12 seconds is approximately 176.186 m/s.
To determine the average speed (rate of climb) of the rocket in the first 12 seconds, we need to calculate the change in altitude during that time period.
Since we know that the rocket experiences a 25% decrease in air pressure, we can assume that the change in altitude is directly proportional to the change in pressure. In other words, the decrease in pressure corresponds to an increase in altitude.
To get the altitude change, we need to know the relationship between pressure and altitude. In standard meteorological conditions, the atmospheric pressure decreases by about 10% for every 100 meters increase in altitude.
Let's assume that the rocket's pressure measuring device is accurate and measures the air pressure at sea level as 100 units (since no specific unit is given). Therefore, a 25% decrease in pressure would result in a pressure reading of 75 units.
Using the relationship between pressure and altitude, we can calculate the altitude change as follows:
Altitude change = (Original pressure - New pressure) / (Original pressure decrease per 100 meters of altitude)
Altitude change = (100 - 75) / (10/100) [substituting the given values]
Altitude change = 25 / 0.1 [performing the calculation]
Altitude change = 250 meters
Therefore, in the first 12 seconds, the rocket has climbed 250 meters.
To calculate the average speed (rate of climb), we divide the altitude change by the time taken:
Average speed = Altitude change / Time
Average speed = 250 m / 12 s
Average speed ≈ 20.83 m/s
Hence, the average speed (rate of climb) of the rocket in the first 12 seconds is approximately 20.83 meters per second.