part 1 of 3: A hawk flies in a horizontal arc of radius 9.9 m at a constant speed of 4.8 m/s. Find its centripetal acceleration. Answer in units of m/s^2.
part 2 of 3: It continues to fly along the same horizontal arc but increases its speed at the rate of 1 m/s^2. Find the magnitude of acceleration under these new conditions. Answer in units of m/s^2.
Part 3 of 3: Find the direction of acceleration relative to the direction of motion under these conditions. Answer between -180 degrees and 180 degrees. Answer in unit of degrees.
(a) Centripetal (or normal)
acceleration is
a(c)= m•v²/R=…
(b) Tangential acceleration
a(τ)= 1 m/s²
a=sqrt{a(c)² +a(τ)²}=...
(c)tanα= a(c)/a(τ)= ..
Part 1 of 3: To find the centripetal acceleration of the hawk flying in a horizontal arc, we can use the formula for centripetal acceleration:
Centripetal acceleration = (v^2) / r,
where v represents the speed of the hawk and r represents the radius of the arc.
Given that the speed of the hawk is 4.8 m/s and the radius of the arc is 9.9 m, we can substitute these values into the formula:
Centripetal acceleration = (4.8 m/s)^2 / 9.9 m = 23.04 m^2/s^2 / 9.9 m = 2.33 m/s^2.
Therefore, the centripetal acceleration of the hawk is 2.33 m/s^2.
Part 2 of 3: To calculate the magnitude of acceleration when the hawk increases its speed at a rate of 1 m/s^2, we need to consider the change in speed.
Given that the initial speed of the hawk is 4.8 m/s and the rate of acceleration is 1 m/s^2, we can determine the final speed with the formula:
Final speed = Initial speed + (Rate of acceleration * Time).
Since we are given the rate of acceleration and not the time, we can find the final speed when the hawk increases its speed by 1 m/s. Let's call the final speed vf.
vf = 4.8 m/s + (1 m/s^2 * t).
To calculate the time it takes for the hawk to increase its speed by 1 m/s, we need to use the equation:
Rate of acceleration = (Change in speed) / Time.
Rearranging the equation, we can solve for time:
Time = (Change in speed) / Rate of acceleration.
Substituting the values, we have:
Time = 1 m/s / 1 m/s^2 = 1 s.
So, the time it takes for the hawk to increase its speed by 1 m/s is 1 second.
Now, to find the final speed:
vf = 4.8 m/s + (1 m/s^2 * 1 s) = 5.8 m/s.
Therefore, the new speed of the hawk is 5.8 m/s.
Since the direction of the motion is unchanged, the magnitude of acceleration remains the same. Therefore, the magnitude of acceleration under these new conditions is still 2.33 m/s^2.
Part 3 of 3: To determine the direction of acceleration relative to the direction of motion, we need to consider the vector nature of acceleration.
When an object increases its speed, the acceleration is in the same direction as its motion. Therefore, in this case, the direction of acceleration is the same as the direction of motion, which is considered positive.
Hence, the direction of acceleration relative to the direction of motion is 0 degrees.