To solve this problem, we can use the kinematic equations for free fall under the influence of gravity, assuming no air resistance. The equation for the distance fallen due to gravity is:
\[ d = \frac{1}{2} g t^2 \]
where:
- \( d \) is the distance fallen (height of fall in meters),
- \( g \) is the acceleration due to gravity (approximately 9.8 m/s²),
- \( t \) is the time in seconds.
We know from the data that both rocks are dropped from the same height (78.4 m). We can use the time it takes for Rock 1 to fall to find the acceleration due to gravity and check if it matches with Rock 2.
For Rock 1: \[ 78.4 = \frac{1}{2} (9.8) (4.0)^2 \]
Calculating the right side: \[ 78.4 = \frac{1}{2} \times 9.8 \times 16 \] \[ 78.4 = 78.4 \]
So, the calculations confirm that Rock 1's time of fall is accurate.
Now, we can apply the same calculations to Rock 2. Using the same height of 78.4 m and solving for time \( t \):
\[ 78.4 = \frac{1}{2} (9.8) t^2 \]
Solving for \( t \): \[ 78.4 = 4.9 t^2 \] \[ t^2 = \frac{78.4}{4.9} \] \[ t^2 = 16 \] \[ t = 4.0 , \text{s} \]
Thus, Rock 2 will also take 4.0 seconds to fall.
So, the answer is:
4.0 seconds.