Questions: A peregrine falcon dives at a pigeon. The falcon starts downward from rest with free-fall acceleration. If the pigeon is 76.0 m below the initial position of the falcon, how long does the falcon take to reach the pigeon? a. 3.9 s b. 15.5 s c. 2.8 s d. 7.8 s

A peregrine falcon dives at a pigeon. The falcon starts downward from rest with free-fall acceleration. If the pigeon is 76.0 m below the initial position of the falcon, how long does the falcon take to reach the pigeon?
a. 3.9 s
b. 15.5 s
c. 2.8 s
d. 7.8 s

Transcript text: A peregrine falcon dives at a pigeon. The falcon starts downward from rest with free-fall acceleration. If the pigeon is 76.0 m below the initial position of the falcon, how long does the falcon take to reach the pigeon? a. 3.9 s b. 15.5 s c. 2.8 s d. 7.8 s

Solution

Solution Steps

Step 1: Identify the Known Values

The problem states that the falcon starts from rest and dives with free-fall acceleration. The distance to the pigeon is 76.0 meters. The acceleration due to gravity is \( g = 9.81 \, \text{m/s}^2 \).

Step 2: Use the Kinematic Equation

We use the kinematic equation for motion under constant acceleration: \[ d = v_i t + \frac{1}{2} a t^2 \] where:

\( d = 76.0 \, \text{m} \) (distance)
\( v_i = 0 \, \text{m/s} \) (initial velocity)
\( a = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)
\( t \) is the time we need to find.

Step 3: Simplify the Equation

Since the initial velocity \( v_i = 0 \), the equation simplifies to: \[ 76.0 = \frac{1}{2} \times 9.81 \times t^2 \]

Step 4: Solve for Time \( t \)

Rearrange the equation to solve for \( t^2 \): \[ t^2 = \frac{2 \times 76.0}{9.81} \] \[ t^2 = \frac{152.0}{9.81} \] \[ t^2 \approx 15.4934 \] Taking the square root of both sides gives: \[ t \approx \sqrt{15.4934} \approx 3.9369 \]