Peregrine falcons dive practically straight down and can reach speeds of 100 m/s by the time they reach their prey (smaller birds).
If all of the falcon's speed is coming from its initial potential energy, how high must the falcon be above its prey at the beginning of the dive in order to reach 100 m/s when it catches its prey?
Mass of an average male peregrine falcon in 0.91 kg
A) 7.0 meters
B) 55 meters
C) 510 meters
since PE = KE,
mgh = 1/2 mv^2
h = v^2/(2g)
To find the answer, we need to apply the principle of conservation of energy. At the beginning of the dive, the falcon has only potential energy, which is converted into kinetic energy as it dives and gains speed.
The potential energy of an object is given by the equation:
PE = mgh
Where:
PE is the potential energy
m is the mass of the object
g is the acceleration due to gravity
h is the height above the reference point
In this case, the potential energy at the beginning of the dive is equal to the mass of the falcon multiplied by the acceleration due to gravity multiplied by the height of the falcon above its prey:
PE = mgΔh
Where:
Δh is the height difference between the falcon and its prey
The kinetic energy of an object is given by the equation:
KE = (1/2)mv^2
Where:
KE is the kinetic energy
m is the mass of the object
v is the velocity of the object
At the end of the dive, when the falcon reaches its prey, all the potential energy is converted into kinetic energy. Therefore, we can equate the potential energy at the beginning with the kinetic energy at the end:
mgΔh = (1/2)mv^2
The mass cancels out, and the equation becomes:
gΔh = (1/2)v^2
We can rearrange the equation to solve for Δh:
Δh = (1/2)(v^2)/g
Plugging in the values given:
Δh = (1/2)(100^2)/(9.8)
Δh = 5000/9.8
Δh ≈ 510 meters
Therefore, the falcon must be approximately 510 meters above its prey at the beginning of the dive in order to reach a speed of 100 m/s when it catches its prey.
So, the correct answer is:
C) 510 meters