A rock is dropped from a sea cliff, and the sound of it striking the ocean is heard 3.2 seconds later. If the speed of sound is 340 meters per second, how high is the cliff?
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To determine the height of the cliff, we first need to calculate the time it takes for the sound to travel from the top of the cliff to the bottom.
Given:
- Speed of sound: 340 meters per second
- Time elapsed until the sound is heard: 3.2 seconds
Since the sound travels from the top to the bottom, the time it takes is equal to the time elapsed minus the time it takes for the rock to fall.
Let's denote:
- Time for the rock to fall: t
So, the time for the sound to travel downward from the top of the cliff to the ocean is 3.2 seconds - t.
Now, let's calculate the time it takes for the rock to fall using the formula:
Distance = (1/2) * g * t^2
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Assuming the rock is dropped (not thrown), its initial velocity is 0 m/s.
We want to find the time it takes for the rock to fall, so we can rearrange the formula as follows:
t = sqrt((2 * Distance) / g)
Since the height of the cliff is the same as the distance the rock falls, we can substitute Distance with the height of the cliff.
t = sqrt((2 * Height) / g)
Now, we have an equation to determine the time it takes for the rock to fall.
Next, we can substitute this time value into the equation we derived earlier:
3.2 seconds - t = time for sound to travel downward
To solve for the height, we need to equate the time it takes for the rock to fall to the time it takes for the sound to travel.
3.2 seconds - t = t
Now, we can solve for t by rearranging the above equation:
3.2 seconds = 2t
t = 3.2/2
t = 1.6 seconds
Now, we can substitute this t value into the equation for time of fall:
1.6 seconds = sqrt((2 * Height) / g)
To solve for the height, we isolate it:
(1.6 seconds)^2 * g = 2 * Height
Height = (1.6 seconds)^2 * g / 2
Finally, we substitute the values for g and solve for the height:
Height = (1.6 seconds)^2 * 9.8 m/s^2 / 2
Height = 1.6^2 * 9.8 / 2
Height = 4.096 * 9.8 / 2
Height = 40.192 / 2
Height ≈ 20.1 meters
Therefore, the height of the cliff is approximately 20.1 meters.
To determine the height of the cliff, we can use the equation of motion for the falling rock. The formula is:
h = (1/2) * g * t^2
Where:
- h is the height of the cliff
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time it takes for the rock to fall (3.2 seconds)
So, let's calculate the height of the cliff step by step:
Step 1: Calculate the time it takes for the sound to reach the observer on the cliff.
Given that the speed of sound is 340 meters per second, and it takes 3.2 seconds for the sound to reach the observer, the distance covered by the sound can be calculated as:
Distance = Speed * Time
Distance = 340 m/s * 3.2 s
Distance = 1088 meters
Therefore, the sound covered a distance of 1088 meters.
Step 2: Calculate the time it takes for the rock to fall.
Since the sound takes 3.2 seconds to reach the observer, we need to find the total time it took for the rock to fall to the water's surface and the sound to reach the observer. Thus, the time taken by the rock to fall can be calculated as:
Total time = Time for sound + Time for rock
Total time = 3.2 seconds
Step 3: Calculate the height of the cliff.
Using the equation h = (1/2) * g * t^2, we can rearrange the equation to solve for h:
h = (1/2) * g * t^2
h = (1/2) * 9.8 m/s^2 * (3.2 seconds)^2
h = 1/2 * 9.8 m/s^2 * 10.24 s^2
h = 49.96 meters
Therefore, the height of the cliff is approximately 49.96 meters.