In a hurricane, the wind pressure varies directly as the square of the wind velocity. If wind pressure is a measure of a hurricane’s destructive capacity, what happens to this destructive power when the wind speed doubles?
Do you have an equation to prove if this makes sense of not?
p = kv^2
so, if v is replaced by 2v, then p becomes
k(2v)^2 = k*4v^2 = 4kv^2 = 4p
So, p grows by a factor of 4. Makes sense, since p grows as v^2.
To determine what happens to the destructive power of a hurricane when the wind speed doubles, we can use the equation relating wind pressure to wind velocity. According to the given information, wind pressure varies directly as the square of the wind velocity.
Let's represent wind pressure as P and wind velocity as V. The equation would then be written as:
P = kV^2
Here, k represents the constant of proportionality.
Now, if we double the wind speed to 2V, we can substitute this into the equation:
P = k(2V)^2
P = k(4V^2)
P = 4kV^2
Comparing this to the original equation, we can see that when the wind speed doubles, the wind pressure (and hence the destructive power) increases by a factor of 4. Hence, doubling the wind speed results in a fourfold increase in the destructive power of the hurricane.
By using the equation, we can mathematically prove that this relationship makes sense.
To determine what happens to the destructive power of a hurricane when the wind speed doubles, let's first understand the relationship between wind pressure and wind velocity based on the given information.
The problem states that wind pressure varies directly as the square of the wind velocity. This means that the wind pressure (P) is proportional to the square of the wind velocity (V), or mathematically:
P ∝ V^2
Now, to find the equation relating wind pressure and wind velocity, we need to introduce a constant of proportionality. Let's call this constant k:
P = kV^2
To determine if this equation makes sense, we can analyze what happens when the wind speed doubles. Doubling the wind speed means multiplying the original wind speed by 2. Let's denote the original wind speed as V1 and the doubled wind speed as V2.
From the equation, we can compare two scenarios:
Scenario 1: P1 = kV1^2
Scenario 2: P2 = kV2^2
Since V2 is twice as large as V1, we know that V2 = 2V1. Substituting this into Scenario 2:
P2 = k(2V1)^2
P2 = k(4V1^2)
P2 = 4kV1^2
Comparing P1 and P2, we can see that P2 is four times as large as P1 when the wind speed doubles. Therefore, doubling the wind speed results in a four-fold increase in the destructive power of the hurricane, as measured by wind pressure.
In conclusion, when the wind speed doubles, the destructive power of the hurricane, as measured by wind pressure, increases by a factor of four.
You can use this equation and method to determine how the destructive power changes for different wind speed ratios in similar scenarios.