Final answer:
Birdman should fly at a speed of approximately 6.7056 m/s to hit the bucket placed 123 m from the start of the field while flying at a height of 90 m.
Step-by-step explanation:
To determine how fast Birdman should fly to hit the bucket placed 123 m from the start of the field while flying at a height of 90 m, we can use the concept of projectile motion.
In projectile motion, the horizontal and vertical motions are independent of each other. The horizontal motion is at a constant speed, while the vertical motion is influenced by the acceleration due to gravity.
To hit the bucket, Birdman needs to reach the horizontal distance of 123 m while maintaining a height of 90 m.
Let's assume the speed at which Birdman flies is v m/s.
The time taken to reach the bucket horizontally can be calculated using the equation:
time = distance / speed
Therefore, the time taken to cover the horizontal distance of 123 m is:
time = 123 m / v m/s
During this time, the vertical motion is affected by the acceleration due to gravity. The vertical motion can be described using the equation of motion:
vertical displacement = initial vertical velocity * time + (1/2) * acceleration due to gravity * (time)^2
Since Birdman is flying at a constant height of 90 m, the vertical displacement is 0 m. We can substitute the known values into the equation:
0 m = 90 m/s * time + (1/2) * (-9.8 m/s^2) * (time)^2
Simplifying the equation, we get:
0 = 90t - 4.9t^2
Rearranging the equation, we have:
4.9t^2 - 90t = 0
Factoring out a t, we get:
t(4.9t - 90) = 0
From this equation, we have two possible solutions:
1) t = 0
2) 4.9t - 90 = 0
Since the time cannot be zero (t = 0), we consider the second equation:
4.9t - 90 = 0
Solving for t, we find:
4.9t = 90
t = 90 / 4.9
Now, we can substitute the value of t into the equation for the horizontal distance:
time = 123 m / v m/s
90 / 4.9 = 123 / v
Cross multiplying, we get:
90v = 123 * 4.9
Simplifying, we have:
v = (123 * 4.9) / 90
Therefore, Birdman should fly at a speed of approximately 6.7056 m/s to hit the bucket placed 123 m from the start of the field while flying at a height of 90 m.