A hawk flies in a horizontal arc of radius 12.0 m at a constant speed of 7.1 m/s. It continues to fly along the same horizontal arc but increases its speed at the rate of 7.9 m/s2. Find the acceleration magnitude under these conditions.
radial A = v^2/r
tangential A = 7.9
they are perpendicular
|A| = sqrt ( v^4/r^2 + 7.9^2 )
To find the acceleration magnitude of the hawk under these conditions, we need to consider the two types of acceleration involved: centripetal acceleration and tangential acceleration.
First, let's find the centripetal acceleration. Centripetal acceleration is given by the formula:
a = v^2 / r
Where:
a is the centripetal acceleration
v is the velocity (speed) of the hawk
r is the radius of the arc
In this case, the velocity of the hawk is given as 7.1 m/s, and the radius of the arc is 12.0 m. Substituting these values into the formula, we get:
a = (7.1 m/s)^2 / 12.0 m
a = 0.272 m/s^2
Now, let's find the tangential acceleration. Tangential acceleration is the rate at which the speed of an object changes. In this case, the speed of the hawk is increasing at a constant rate of 7.9 m/s^2.
Therefore, the magnitude of the tangential acceleration is simply the given rate of change of speed:
at = 7.9 m/s^2
To find the total acceleration magnitude, we need to combine the centripetal acceleration and tangential acceleration. Since they act in perpendicular directions, we can use the Pythagorean theorem to find the magnitude of their vector sum:
acceleration magnitude (a) = √(ac^2 + at^2)
Substituting the values we found earlier:
a = √((0.272 m/s^2)^2 + (7.9 m/s^2)^2)
a ≈ 7.9 m/s^2
Therefore, the acceleration magnitude of the hawk under these conditions is approximately 7.9 m/s^2.