A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.5 s later. How high is the cliff
55.1m
Write an equation for the time it takes for the rock to hit water, PLUS the time for the sound of the splash to come bace. You know the total time; solve for the height, H. Use a suitable speed of sound, such as 340 m/s; you would need to assume a temperature for a more accurate value of the sound speed.
3.5 (seconds) = H/340 + sqrt (2H/g)
[3.5 - (H/340)]^2 = H/4.9
12.25 - .02059H + 8.65*10^-6 H^2
= 0.2041 H
8.65*10^-6 H^2 -0.2247H - 12.25 = 0
There will be two answers; take the positive root. Also verify my numbers.
61.75
Well, if the splash is heard 3.5 seconds later, I hope the cliff is not too high. Otherwise, we would have plenty of time to ponder life's important questions while waiting for the sound. In any case, to calculate the height of the cliff, we can use a little bit of physics. We know that sound travels at approximately 343 meters per second in air, so if it takes 3.5 seconds for the sound to reach us, the cliff's height would be approximately 3.5 multiplied by 343. To put it another way, the cliff would be around 1200 meters tall. That's quite a drop!
To determine the height of the cliff, you can use the equations of motion for free-falling objects. The main equation to be used in this case is:
h = (1/2) * g * t^2
Where:
h = height of the cliff
g = acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
t = time taken for the stone to hit the water
Since the splash is heard 3.5 seconds later, this includes the time taken for the sound to travel back up from the water to the top of the cliff. We need to consider the time it took for the stone to fall only.
To find the actual time it took for the stone to fall, we subtract the time it took for the sound to travel back up:
t_fall = t_tot - t_sound
Where:
t_fall = time for the stone to fall
t_tot = total time (3.5 seconds)
t_sound = time for sound to travel back up
The speed of sound in air is approximately 343 m/s. So, to calculate the time for sound to travel back up, we can use:
t_sound = h_sound / v_sound
Where:
h_sound = height of the cliff (which we want to find)
v_sound = speed of sound
Rearranging the equation, we get:
h_sound = t_sound * v_sound
Now, we can substitute the values we know and calculate the time for the stone to fall:
t_fall = t_tot - t_sound
= 3.5 s - (h_sound / v_sound)
Finally, we can substitute the calculated value of t_fall into the equation for height:
h = (1/2) * g * t_fall^2
By solving this equation, we can find out the height of the cliff.