A hawk flies in a horizontal arc of radius 16 m at a constant speed of 4.8 m/s. Find its centripetal acceleration.
Answer in units of m/s2. It continues to fly along the same horizontal arc but increases its speed at the rate of 1.52 m/s2. Find the magnitude of acceleration under these new conditions. Answer in units of m/s2.
Find the direction of acceleration relative to the direction of motion under these conditions. Answer between −180◦ and 180◦. Answer in units of ◦.
To find the centripetal acceleration of the hawk in the first scenario, we can use the formula:
ac = v^2 / r
where ac is the centripetal acceleration, v is the velocity, and r is the radius.
Substituting the given values into the formula, we have:
ac = (4.8 m/s)^2 / 16 m = 1.44 m/s^2
So, the centripetal acceleration of the hawk is 1.44 m/s^2.
In the second scenario, the acceleration of the hawk is given as 1.52 m/s^2, which is the rate at which its speed increases. The magnitude of acceleration, in this case, is equal to the tangential acceleration.
The tangential acceleration can be calculated using the formula:
at = dv / dt
where at is the tangential acceleration, dv is the change in velocity, and dt is the change in time.
Since the speed of the hawk is increasing uniformly, we can assume the acceleration is constant.
Using the given acceleration and using the equation at = a, we can equate the tangential acceleration to the given value:
at = 1.52 m/s^2
By rearranging the formula, we have:
dv = at * dt
By integrating both sides of the equation, we get:
Δv = at * Δt
where Δv is the change in velocity and Δt is the change in time.
Now we know that velocity is the product of speed and direction. Since the hawk is flying in a horizontal arc, the direction of acceleration is perpendicular to the direction of motion, which means it is perpendicular to the velocity vector.
Therefore, the direction of acceleration relative to the direction of motion is at a right angle or 90 degrees.
To find the centripetal acceleration of the hawk, we'll use the formula:
Centripetal acceleration (ac) = (velocity squared) / radius
First, let's calculate the centripetal acceleration of the hawk when it is flying at a constant speed of 4.8 m/s.
Given:
Velocity (v) = 4.8 m/s
Radius (r) = 16 m
Using the formula, we have:
ac = (4.8^2) / 16
Calculating:
ac = 23.04 / 16
ac ≈ 1.44 m/s^2
So, the centripetal acceleration of the hawk when it is flying at a constant speed is approximately 1.44 m/s^2.
Next, let's find the magnitude of acceleration when the hawk increases its speed at a rate of 1.52 m/s^2. In this case, we need to consider the rate of change of speed as well.
Given:
Rate of change of speed (a) = 1.52 m/s^2
To find the magnitude of acceleration, we'll add the centripetal acceleration (ac) to the rate of change of speed (a):
Magnitude of acceleration = ac + a
Substituting the known values:
Magnitude of acceleration = 1.44 + 1.52
Magnitude of acceleration = 2.96 m/s^2
So, the magnitude of acceleration of the hawk under these new conditions is 2.96 m/s^2.
Finally, let's find the direction of acceleration relative to the direction of motion. In this case, the hawk is experiencing an increase in speed, so the direction of acceleration will be in the same direction as the velocity.
The direction of acceleration relative to the direction of motion is 0° or in the same direction as the hawk's velocity.
Therefore, the direction of acceleration relative to the direction of motion is 0°.
a=v^2/r
=4.8^2/16
=23.04/16
a=1.44