the peregrine falcon is the world's fastest known bird and has been clocked driving downward toward its pray at a constant vertical velocity of 97.2 meters per second. If the falcon drives straight down from a height of 100 meters, how much time does this give a rabbit below to consider his next move as the falcon begins his decent?
To find the time it takes for the falcon to reach the ground, we can use the equation of motion:
\[s = ut + \frac{1}{2}a t^2\]
where:
- \(s\) is the distance traveled (100 meters in this case)
- \(u\) is the initial velocity (97.2 meters per second)
- \(a\) is the acceleration (in this case, it is the acceleration due to gravity, approximately 9.8 m/s^2)
- \(t\) is the time taken
Rearranging the equation, we get:
\[\frac{1}{2} a t^2 + u t - s = 0\]
Substituting the known values:
\[\frac{1}{2} \times 9.8 \times t^2 + 97.2 \times t - 100 = 0\]
Now, we can solve this quadratic equation for \(t\). Let's calculate it using the quadratic formula:
\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this case, \(a = \frac{1}{2} \times 9.8\), \(b = 97.2\), and \(c = -100\).
\[t = \frac{-97.2 \pm \sqrt{(97.2)^2 - 4 \times \frac{1}{2} \times 9.8 \times -100}}{2 \times \frac{1}{2} \times 9.8}\]
Calculating the value within the square root:
\[\sqrt{(97.2)^2 - 4 \times \frac{1}{2} \times 9.8 \times -100} = \sqrt{9464.64 + 1960} = \sqrt{11424.64} \approx 106.86\]
Simplifying the equation further:
\[t = \frac{-97.2 \pm 106.86}{9.8}\]
Solving for both cases (the positive and negative sign):
Case 1: \(t = \frac{-97.2 + 106.86}{9.8} = \frac{9.66}{9.8} \approx 0.9865\) seconds
Case 2: \(t = \frac{-97.2 - 106.86}{9.8} = \frac{-204.06}{9.8} \approx -20.82\) seconds
Since time cannot be negative in this context, we conclude that the positive value of 0.9865 seconds is the time it takes for the falcon to reach the ground.
Therefore, the rabbit has approximately 0.9865 seconds to consider its next move as the falcon begins its descent.