A hawk flies in a horizontal arc of radius 12.0 m at a constant speed of 5.4 m/s. It continues to fly along the same horizontal arc but increases its speed at the rate of 2.8 m/s2. Find the acceleration magnitude under these conditions.

v^2/R radial

and
2 tangential

a = sqrt( [v^2/R]^2 + 2.8^2 )

but v = 5.4 + 2.8 t

To find the acceleration magnitude, we need to calculate the magnitude of the acceleration when the hawk is increasing its speed.

The acceleration of an object moving in a circular path can be calculated using the formula:

a = v^2 / r

Where:
- a is the acceleration
- v is the velocity or speed
- r is the radius of the circular path

In this case, the hawk is flying in a circular path of radius 12.0 m, and its initial speed is 5.4 m/s. Therefore, the initial acceleration is given by:

a_1 = (5.4 m/s)^2 / 12.0 m

Simplifying the equation gives:

a_1 = 2.43 m^2/s^2

Next, the hawk increases its speed at a rate of 2.8 m/s^2. To find the acceleration magnitude when the hawk is increasing its speed, we need to add this rate of change to the initial acceleration:

a_2 = a_1 + 2.8 m/s^2

Substituting the value of a_1, we get:

a_2 = 2.43 m^2/s^2 + 2.8 m/s^2

Simplifying the equation gives:

a_2 = 5.23 m^2/s^2

Therefore, the acceleration magnitude under these conditions is 5.23 m^2/s^2.

To find the acceleration magnitude, we need to evaluate the magnitude of the hawk's acceleration when it increases its speed at a rate of 2.8 m/s².

Since the hawk is flying in a horizontal arc, its acceleration will be directed towards the center of the arc, providing the necessary centripetal force to keep it moving along the circular path. The magnitude of this centripetal acceleration can be calculated using the formula:

a = v² / r

Where:
- a is the acceleration magnitude
- v is the velocity of the hawk
- r is the radius of the arc

In the initial scenario, where the hawk flies at a constant speed of 5.4 m/s, we can substitute the given values into the formula:

a₁ = (5.4 m/s)² / 12.0 m

Now, let's calculate the value:

a₁ = (29.16 m²/s²) / 12.0 m
a₁ = 2.43 m/s²

So, in the initial scenario, the magnitude of the hawk's acceleration is 2.43 m/s².

In the second scenario, where the hawk increases its speed at a rate of 2.8 m/s², we need to find the new magnitude of the acceleration.

First, we need to find the new speed by adding the acceleration rate over time:

v₂ = v₁ + (a × t)

Since the hawk's initial speed is 5.4 m/s and its acceleration rate is 2.8 m/s², we can substitute these values into the equation:

v₂ = 5.4 m/s + (2.8 m/s² × t)

Next, we need to find the time it takes for the speed to increase. Since no time is given in the question, we cannot find the exact magnitude of the acceleration. We can only express it in terms of the rate of acceleration.

So, the magnitude of the acceleration under these conditions cannot be determined without knowing the time period over which the speed increases.