Lb T.

asked • 02/12/17

A stone falls from the top of a cliff into the ocean.

A stone falls from the top of a cliff into the ocean.
In the air, it had an average speed of 16 m/s. In the water, it had an average speed of 3 m/s before hitting the seabed. The total distance from the top of the cliff to the seabed is 127 meters, and the stone's entire fall took 12 seconds.
How long did the stone fall in air and how long did it fall in the water?

Jake C.

I used a guess and check method in my head. My answer will be two separate numbers that add up to 12. Because we have 12 seconds to fall in total and two separate periods of falling. In terms of ratios between the numbers. The divisibility of 127 strikes out at me as something that won't work cleanly for 16. but would be easier for 3 potentially if I chunk the number up a bit. In other words we have 127 feet to split between multiples of 3 and 16. As I already mentioned it is immediately clear to me that 127 is not divisible by 16 but the number 27 is divisible by 3 and that would leave a 100 leftover from 127. Sadly 100 is not divisible by 16. Another reason it can't be 9 seconds falling at 3 m/s is because we have a whole 127 feet to fall. We need a better way to split our 12 seconds into two numbers that add up to 12 seconds in which the 16 m/s or in the air time will be bigger than the 3m/s in the water time. Unfortunately a split of 9 seconds and 3 seconds doesn't work so let's continue. So as a thought experiment to further drive home this point let's try and let the object travel as slow as 3 meters per second the whole time. In doing so it won't have nearly enough time to cover the full 127 feet. For example if we assumed it fell the whole time at 3m/s instead it would take 127/3 seconds or 42 seconds (that is of course way too long). We only have 12 seconds in total to share so that can't be right. What about checking multiples of 16 up to 127. So we have 16, 32, 48, 64, 80 96, 112, 128. STOP that is over 127. So if we do 8*16 we go over the feet needed. Because 128 is greater than 127. So 16*7=112 is a good spot to work with. Where 16 is the number of feet and 7 is the number of seconds. Okay this seems promising. 12-7 = 5 seconds left over. Also 127 feet - 112 feet is 15 feet leftover. Lets now recall the slower speed the object travels at. It was 3 meters per second. Lucky for us the 15 feet leftover is easily divisible by 3 m/s since 15/3=5. Looks like we solved it. We managed to find the right answer. 7 seconds in the air and 5 seconds in the water.
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11/12/23

2 Answers By Expert Tutors

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Michael C. answered • 02/12/17

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PhD with 20 years experience teaching math and applied science

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total distance = sum of velocity * time for both legs of the journey;
 
let the time taken in air be t
 
hence time taken in water is 12-t
 
127 = 16*t + 3(12-t)
 
127 = 16t + 36 - 3t
 
127 - 36 = 16t - 3t
 
91 = 13t
 
t = 91/13 = 7 seconds 
 
Hence time taken in air = 7 seconds
 
time taken in water is 12 - 7 = 5 seconds
 
checking; 16*7 + 3*5 = 127 (proven)
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