Now Math(s)/Physic(s) are not my strong point, so please bear with me! On holiday we found, in the grounds of a stately home, a pit which was originally constructed to keep freshly-caught fish from decaying, by being covered with ice collected in winter time. The entrance is now barred for reasons of safety. I dropped several small pebbles through the bars into it to try to determine its depth. As near as I could calculate the time was exactly 3/4 of one second from releasing each stone to its landing (not very deep, I know, but what the hell!) How deep was the pit, and how is the formula of 32ft/second squared employed? I would guess at roughly 7 feet, but that is hardly scientific! The time, though, is pretty accurate, I would say.

The formula 32 feet/sec^2 refers to the acceleration, which is constant for a falling object if you ignore air resistance. Thus after one second its speed is 32 ft/sec. After two seconds its speed is 64 ft/sec, and so on. It picks up speed (accelerates) at the rate of 32 ft/sec per second = 32 ft/sec^2. OK so far?

To convert elapsed time into a total distance fallen, however, is tricky because the speed is constantly changing. This is where calculus flexes its muscles and makes short work of the problem.

Using the methods of calculus, it's easy to show that after time T the distance fallen is (1/2)*A*T^2, where A=acceleration, * denotes multiplication, and T^2 means "T-squared" [I don't know how to do superscripts here].

So with A=32, the formula becomes simply: Distance = 16*T^2. Since T=3/4, we have D=16*(9/16) = 9 feet.

Your guess of 7 feet was a little low -- 22% underestimated. Just don't try to climb in using an 8-foot ladder.

Many thanks, Professor! I half expected Maiku to leap in here, but you got there first, and thanks for a comprehensible (to me!) reply. Just one question; you say:- "Thus after one second its speed is 32 ft/sec. After two seconds its speed is 64 ft/sec, and so on.". Do you mean that after 3 seconds its speed is 128 feet per second, or 96 feet per second? Sorry to be thick, the only scientific national exam I took was mathematics itself when I was 16, and that is a very long time ago...

I will take a ten-foot ladder, just to be on the safe side...

No. After three seconds its speed is 96 ft/sec. At the end of each second it's speed has increased another 32 ft/sec. The speed rises linearly and the distance increases as the square of time.

There is no successive doubling involved, which would be "exponential." Sorry not to have been clear about that. It's especially confusing due to the coincidence that, in English units, the acceleration is 32 which is a power of 2.

Thanks, Prof; got it at last! Clear as day. What is your own speciality? email me if you wish, my address is in my profile.

Just as a matter of interest Ritz, which stately home was this? I remember going to one around the area you went. The 'ice house' was just a hole in the ground.

The wonderful Culzean Castle, Jen. (Pronounced 'Cullain') containing Eisenhower Apartment I know, off-topic! I will post it in 'Travel' now...