No, that last integral was wrong.......
It should be
t= \frac{1}{ \sqrt {2g} } \int_0^{16} \frac{ \sqrt {\frac {1}{4y}+1} }{ \sqrt {(16-y)} } \,\, dy
The following two integrals seem to be equal but not proven yet
t= \frac{1}{ \sqrt {2g} } \int_0^4 \frac{ \sqrt {4x^2+1} }{...
,
The cylindrical slice of width dx gives the exact volume (not an approximation!) of a cylindrical shell. For a sphere, finding a volume first with cylindrical shells then taking a derivative of the volume will give the surface area of the sphere. But for other non symetric volumes that...
My first numerical analysis was wrong. I looked over the equations in my excel spread sheet and found that I had squared a value when it should have been a square root.
That mistake was found when I did an approximation on the above integral - and the numbers did not jive. This gives it a...
One thing for sure, that integral is not easy to solve. I ran it through wolfram integrator and the solution used appell hypergeometric functions.
The obvious thing to do is put the 2g constant outside the integration sign:
t= \frac{1}{ \sqrt {2g} } \int_0^4 \frac{ \sqrt {4x^2+1} }{...
This thread is about variable acceleration so
The following definite integral should calculate the time
t= \int_0^4 \frac{ \sqrt {4x^2+1} }{ \sqrt {2g(16-x^2)} } \,\, dx
I'm not sure if this is correct because I have not yet solved it Any ideas on how to solve that integral?
I've done a bit of numerical analysis on this problem
I used a ramp in the shape of the curve y = x^2 that is 16 feet high and 4 feet wide (some real world numbers)
The mass is released at the top of the ramp and slides to the bottom. The question is how long does it take for the mass to...
A little bit more on that last equation. I find it interesting that the actual bank balance does not exactly match the curve produced by the equation as t varies through all values. Although it matches exactly at the time the deposit is made. (when (t) is at integer values)
I think it is...
I think that your invention has limited use. With all the depth sounding equipment available why not make a remote control diver that maintains a set depth regardless of the towing speed. It would be like an underwater sub that you tow. That would eliminate all the headache of figureing how...
The following equation calculates the future value of a series of deposits where the deposit is made at the start of the time period. It took me about 2 hours of work because I had to solve for a geometric series.
Example you start by depositing an amount (a) into a bank account with zero...
The future value of a continuous annuity is
F = \frac{R}{r} (e^{rt} - 1)
Where R is the continuous rate of deposit (dollars per year(t) )
r is the annual non-compounded interest per dollar. (Also known as the APR)
When will the interest on the Future value (F) equal R (one dollar per...
I want to put all this in one post for reference. I started the ball rolling (no pun intended) with this thread.
https://www.physicsforums.com/showthread.php?t=521136
where I developed the idea of solving the volume of a sphere by trig functions.
Then in another thread, I solved for the...
The integral can be used to find the surface area of a spherical cap
A = 2\pi r^2 \int_0^{a/r} \frac {x}{\sqrt{1-x^2}} \,\, dx
Where "a" is the radius of the spherical cap and "r" is the radius of the sphere. I changed the variable of integration from r to x to allow the 2 pi r^2 in...
Here are some more details
Solve the following indefinite integral and then plug in 2pi and the limits later on.
\int \frac{r}{\sqrt {1-r^2}} \,\,dr
substitute
u = 1-r^2 \,\,\,\,\,\,\,\,\,\,\,\, -\frac{1}{2}du = r\,\,dr
rewrite original integral with u as the new variable...