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## How high is the fort?

A cannon ball leaves a fort with an initial horizontal speed of 1.8*10^2 m/s and strikes a ship in the sea below 7.2 s later. How high is the fort above sea level?

Initial height : Δy=½gt² =½(9.8)(7.2)² = 250 m.

Note that the answer is independent of the initial horizontal velocity.

Hey Sun -- constant g ... use your delta V's of 10 @ sec ... Vdn at sea is 72m/s ...
Vave is 36m/s ... freefall 36m/s for 7.2s ==> 210 +42 +7 ~260m ... Regards :)
This takes a little longer to explain, instead of looking up a strange formula on a
formula sheet or even worst trying to memorize the formulas - but I think it is easier.

You can find the answer using the same smarts you use to figure out how much money
you need to buy 10 candy bars a day for the next 7 days if candy bars are \$10.00 (or \$9.81).

You can use the formulas or think of it this way.

1 - One important thing to remember with falling object  or  X & Y problems
is that the X and Y are independent. Always.
So for this we Only need to solve a Y or falling problem.

2- What is the starting velocity?  Zero - 0
3- After 7.2 seconds falling (when it hit the water) how fast was it going?
7.2 sec. X  10 meter per sec. increase every sec. = 7.2 s X 10 m/s2 = 72 m/s
(if you want to make the problem harder and be a minuscule
2.5% more accurate you can use 9.81 m/s2 instead of 10 m/s2)

4- Okay now you know the starting speed and the ending speed;
what was the average speed?   (0 + 72) ÷ 2 = 36  so an average of 36 m/s

5- Good! Now, if something goes an average of 36 m/s for 7.2 s, how far does it go?
36 m/s for 7.2 s = 36 m/s X 7.2 s = 260 m

6) So it fell 260 m therefor the fort was 260 m above sea level.
(Using 9.81 instead of 10 would give 254   -  we were 2.4 % high)

There is nothing strange about the appropriate kinematic equation for projectile motion problems such as this one. The formula I used is well-known and was derived by combining your steps 2 through 5 into one, for any falling time. Using this equation instead of going through your four steps for every falling time is merely a matter of efficiency.
The formulas are strange to a 1st year high school physics student, and to most people except physics and other science and math people.  It is a a good part of thereason 90% of the people shudder when I get to the word physics in telling them what I do.

I know the formula combines many steps and is the scientifically correct way to express the solution.
And the formula provides the answer in one step - but in a single step that is much more complicated.
I always want my students to understand what is going on, not use a formula sheet.

In this part of the country most schools are going to Physics First where All 9th graders in a school take physics. And a significant number of students I have had were freshmen.  Physics needs to be much more conceptual for them.  And it need to be more conceptual for the older students too who are not going to go the math or science route in higher education.  Not everyone needs to do mathematical physics but everyone should understand physics to be good citizens.  And I have always found even with my Juniors and Seniors, they understand it better if it is 1st explained in simpler terms conceptually and they are given formulas after.
Besides, what's wrong solving a problem by intuition because you understand it rather than looking up formulas that you know work but are not sure why?
I absolutely agree with you as far as understanding physics on a conceptual level first, even though intuition can also easily limit or fail you. (In this case, even the slight generalization of the problem to non-zero angles will require trigonometry). I wish my students would understand the derivations of formulas and the deeper concepts behind them, but due to time limitations that is usually not the case. I also wish students in this country would take some physics every year starting in 7th grade, as is the case where I grew up.

I believe Sun is taking physics at the college level now for the first time and may at some point be confronted with a standardized testing situation (GRE, MCAT etc). Unfortunately, students are expected to get to the final answer as quickly as possible, so they often do end up memorizing a set of formulas. I would hope that Sun has seen the kinematic equations by this point.
The tricky moment in your consideration, Robert, is the application of average velocity. This works only for constant acceleration. If you do not specify this, students may blindly apply the same trick to the variable acceleration.

Second, when you said the starting velocity is zero, it bugged even me for a second. Of course, I quickly realized what you meant. But will the student? You implicitly talked about direction of velocity here.

Third, (not directly related to this problem, but important to kinematic problems) the problem involving decelerated motion is the most difficult, like dropping something from a steadily rising helicopter or rocket travelling up and having its engines fail. Students have to assign appropriate signs to what we would call projections of velocities and accelerations. But they lack math and most do not know vectors well, if at all. And what about general case of projectile motion? This is even more difficult. So I do not see how we could get students to know physics other way than telling them to either memorize formulas or be ready to shed lots of "blood, sweat and tears" in order to learn and understand concepts.
I did not know we could look up students status, so he may very well be a university student.
I was also not considering this being a question of only one student, and trying to give an alternate explanation for others.
If here is just for one student to ask a question; then if there is one correct solution a second is not needed, and I should look from now on who the student is.

Kirill,  You are absolutely right about the 0 starting velocity.  In this problem we are solving only the vertical so I just considered only that.  For a student that needs my explanation to 'get it' they would automatically see what I did, I need to remember to point that out, thanks.

About constant acceleration, that is true; but even the formulas would fail for an acceleration that was not constant. Even in a more mathematical physics the derivations of the formulas are for constant acceleration.  You need to get fairly high in university physics before a non-zero jerk is considered.

You talk about physics in 7th grade (they do have physical science sometimes).  You are definitely right. We do not teach science properly here in the US.  Imagine if in in 9th grade everyone learned all Spanish and took a test at the end of the year.  Then in 10th all French and had a test; German in 11th, and Latin in 12th!  Everyone knows you can't learn a language in a year; but expects kids to learn a whole branch of science.