A rock is dropped from a sea cliff, and the sound of the of it striking the ocean is heard 3.2s later. If the speed of sound is 340m/s, how high is the cliff?
To find the height of the cliff, we can use the equation of motion for free fall:
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
In this case, we need to account for the time it takes for the sound of the rock hitting the ocean to reach the top of the cliff. The sound travels at a speed of 340 m/s.
Let's calculate the time it takes for the rock to fall:
t = 3.2 s
Now, we need to subtract the time it takes for the sound to travel back up to the cliff from the total time:
t_total = t + t_sound
Since the sound travels from the ocean to the cliff, the total time will be:
t_total = t + (2 * distance / speed_of_sound)
We need to solve for the distance, so rearranging the equation:
distance = (speed_of_sound * (t_total - t)) / 2
Plugging in the given values:
distance = (340 m/s * (3.2 s - t)) / 2
Finally, substituting the calculated distance into the equation of motion for free fall:
h = (1/2) * g * t^2 + distance
Simplifying the equation:
h = (4.9 m/s^2 * t^2) + distance
Plugging in the values and calculating:
h = (4.9 m/s^2 * (3.2 s)^2) + [(340 m/s * (3.2 s - t)) / 2]
h = 50.176 m + [(340 m/s * (3.2 s - 3.2 s)) / 2]
h = 50.176 m
Therefore, the height of the cliff is approximately 50.18 meters.
To solve this problem, we need to find the time it takes for the rock to fall and then use that time to calculate the height of the cliff. We can use the equation:
time = distance / speed
Let's break down the given information step-by-step:
Step 1: Determine the time it takes for the rock to fall
We are told that the sound of the rock striking the ocean is heard 3.2 seconds later. This means that the sound traveled for 3.2 seconds. Since sound travels at a speed of 340 m/s, we can calculate the distance traveled by sound using:
distance = time × speed
distance = 3.2 s × 340 m/s
distance = 1088 m
Therefore, the rock took 3.2 seconds to fall and reached a height of 1088 meters.
Step 2: Calculate the height of the cliff
The total time it took for the sound to reach our ears includes both the time it took for the rock to fall and the time it took for the sound to travel back up to us. So the total time can be expressed as:
total time = time it took for the rock to fall + time it took for the sound to travel back up
We know that the time it took for the rock to fall is 3.2 seconds. And since sound travels at a speed of 340 m/s, the time it took for the sound to travel back up is the same as the time it took for the sound to reach us, which is also 3.2 seconds.
Therefore, the total time is:
total time = 3.2 s + 3.2 s
total time = 6.4 s
Now, we can use the formula for distance again to calculate the height of the cliff:
distance = total time × speed
distance = 6.4 s × 340 m/s
distance = 2176 m
Thus, the height of the sea cliff is 2176 meters.
If the height is h, and it takes t seconds to fall, then
h = 4.9t^2
and since time = distance/speed,
h/340 + √(h/4.9) = 3.2
h = 46m