To estimate the height of a bridge, a man drops a stone into the water below. How high is the bridge (a) if the stone hits the water 3 seconds later? (b) if the man hears the splash 3 seconds later? (1080 ft per second for speed of sound)
To estimate the height of the bridge in this scenario, we can use the equations of motion and the information provided.
(a) If the stone hits the water 3 seconds later:
We can use the equation of motion for free fall:
h = 0.5 * g * t^2
Where:
h is the height,
g is the acceleration due to gravity (which is approximately 32.2 ft/s^2),
and t is the time taken.
In this case, since the stone takes 3 seconds to hit the water, we can substitute the values into the equation:
h = 0.5 * 32.2 ft/s^2 * (3 s)^2
h = 0.5 * 32.2 ft/s^2 * 9 s^2
h ≈ 0.5 * 290.7 ft
h ≈ 145.35 ft
Therefore, the estimated height of the bridge is approximately 145.35 feet if the stone hits the water 3 seconds later.
(b) If the man hears the splash 3 seconds later:
We need to consider the speed of sound to estimate the height of the bridge in this case.
At a temperature of about 32°F, the speed of sound is approximately 1080 ft/s.
The total time taken is the time the stone takes to fall plus the time for the sound to travel back up to the man:
Total time = falling time + sound travel time
Given that the stone takes 3 seconds to hit the water, we can calculate the sound travel time:
sound travel time = total time - falling time
sound travel time = 3 s - 3 s
sound travel time = 0 s
Since the sound travel time is 0 seconds, it means the sound reaches the man instantly. Therefore, the time for the sound to travel back up is negligible in this scenario.
So, the height of the bridge is directly proportional to the time taken for the stone to fall. Using the equation of motion mentioned earlier:
h = 0.5 * g * t^2
Substituting the values:
h = 0.5 * 32.2 ft/s^2 * (3 s)^2
h = 0.5 * 32.2 ft/s^2 * 9 s^2
h ≈ 0.5 * 290.7 ft
h ≈ 145.35 ft
Therefore, the estimated height of the bridge is approximately 145.35 feet if the man hears the splash 3 seconds later.