Theoretical Calculus question...

[StintedVisions]

So I just have a general question about calculus. I learned that if you take the derivative of the function of an objects position in relation to time you find the velocity of that object in relation to time.
Further more if you set the function of the velocity to zero you can find the time that the object reaches it's maximum height. But if the velocity is zero, couldn't that also be the case before the object is hurled into the air or after the object hits the ground (provided there is no bounce to the object... say it just splats on the ground and remains motionless) and not just the time at which it reaches it's maximum height. Perhaps I'm just, as we used to say in the Navy (and now still do), nuking the question, but I mean is that not the case?

Hurray learning stuff.

 

[Deleted member 4993]

So I just have a general question about calculus. I learned that if you take the derivative of the function of an objects position in relation to time you find the velocity of that object in relation to time.
Further more if you set the function of the velocity to zero you can find the time that the object reaches it's maximum height. But if the velocity is zero, couldn't that also be the case before the object is hurled into the air or after the object hits the ground (provided there is no bounce to the object... say it just splats on the ground and remains motionless) and not just the time at which it reaches it's maximum height.

Right before going splat - it is going with considerable velocity.

Then "ground" happens - the way it happens is that ground applies a force on to the object to go splat.

So there are two situations:

1) the object gets out of my hand with some speed upward

- gravity starts to pull the object down reduces its velocity and it goes to zero at the highest point

- then the object turns around and gravity starts to pull on it - the velocity starts to increase

up untill now nothing sudden happened and mathematically we call the "position function" is continuous

2) then "ground" happens - and the object suddenly stopped by another force

The displacement function is not continuous between the moment before the "impact" (splat) and moment after "impact".

We cannot define derivative of a function at the point of discontinuity.

Velocity can become zero because we apply force (Newton's Laws) - at the moment of application of sudden force (impact) the functions become discontinuous and derivative is not defined.



Perhaps I'm just, as we used to say in the Navy (and now still do), nuking the question, but I mean is that not the case?

Hurray learning stuff.

.

 

[JeffM]

So I just have a general question about calculus. I learned that if you take the derivative of the function of an objects position in relation to time you find the velocity of that object in relation to time.
Further more if you set the function of the velocity to zero you can find the time that the object reaches it's maximum height. But if the velocity is zero, couldn't that also be the case before the object is hurled into the air or after the object hits the ground (provided there is no bounce to the object... say it just splats on the ground and remains motionless) and not just the time at which it reaches it's maximum height. Perhaps I'm just, as we used to say in the Navy (and now still do), nuking the question, but I mean is that not the case?

Hurray learning stuff.

As always, Subhotosh Khan gave a wonderful answer, and he knows physics, of which I am horribly ignorant.

I am going to give a more conceptual answer. Differential calculus deals with differentiable functions. (Duh, saying the same thing in only slightly different words.) What that means is that it deals with things that are changing "smoothly." The ball once thrown is changing its position smoothly until it hits the ground. Once it is on the ground it is changing so smoothly that it does not change at all. However, at the instant when the ball hits the ground and goes splat, things are not changing smoothly. At such instants, differential calculus does not apply. (Or it may apply, but in a more complicated way. Subhotosh Khan may be able to give you the physics of going splat, which will probably involve calculus using a different set of functions.)

I am not sure where you are in your studying, but they may have fussed at you about continuous functions. Wherever a function is not continuous, it is not differentiable. (Unfortunately, a function may be continuous and still not be differentiable.) Continuity is a necessary but not sufficient condition of a function being "smooth enough."

 

[StintedVisions]

Thank you for the explanations, I believe I understand. So basically, before and after "flight" of the object the function is discontinuous and therefore can't be derived?

I'm only in Calc I, and we're starting to wrap up the last chapter (second to last test this coming Friday and final three weeks from then). Unfortunately I'm not doing so well in the class, but fortunately I don't think I'm alone in this from the average test scores are pretty bad. Mine however, have been exceptionally dismal. I am in a position of where I can salvage my grade through a lot of hard work and determination as he is curving the final grade of the class.

I actually take Calc I, the semester prior to this but had to drop due to financial problems with financial aid and my veteran benefits and get a job. I did take the first test though and got a strong B, this time around though (at a different school) things are much different.

Unfortunately I found this forum a bit late in the semester but I'm going to try and get past this class and focus more next semester.

 

[JeffM]

Thank you for the explanations, I believe I understand. So basically, before and after "flight" of the object the function is discontinuous and therefore can't be derived?

I'm only in Calc I, and we're starting to wrap up the last chapter (second to last test this coming Friday and final three weeks from then). Unfortunately I'm not doing so well in the class, but fortunately I don't think I'm alone in this from the average test scores are pretty bad. Mine however, have been exceptionally dismal. I am in a position of where I can salvage my grade through a lot of hard work and determination as he is curving the final grade of the class.

I actually take Calc I, the semester prior to this but had to drop due to financial problems with financial aid and my veteran benefits and get a job. I did take the first test though and got a strong B, this time around though (at a different school) things are much different.

Unfortunately I found this forum a bit late in the semester but I'm going to try and get past this class and focus more next semester.

Technically, the position of the ball before, during, and after the flight is a continuous function of time. Given any time, the position of the ball can be determined, and that position is always "close" to the position that it was in a very small increment of time earlier or later. That is an intuitive but not rigorous definition of "continuous." Make sense?

There are times, however, when that position function is not differentiable. Let's think of one of those times as being the splat moment. Before the splat, the ball is traveling downward and its speed is increasing. It has increasingly negative velocity, and then it comes to an abrupt halt. That is a discontinuity. It is velocity as a function of time that is discontinuous, not position. But the velocity function is the derivative of the position function. After the splat, the velocity is zero. Before the splat, the velocity is increasingly negative. The behavior of the velocity changes suddenly at the splat moment; it does jump. So because the velocity function is discontinuous at the splat moment, we say that the position function is not differentiable at the splat moment.

The difference is this. If a function is continuous at point x, the value of the function at point x is very close to the value of that function at points very close to point x. That is not hard to understand. Intuitively, it means there are no sudden jumps in value. If a function is differentiable at point x, it is continuous at point x, but it has another feature. The changes in the value of a function differentiable at x between two points that are both very close to x and are very close to each other, those changes are equal or almost equal. To be super-informal, a continuous function is smooth, but a differentiable function is extra smooth. Differential calculus is the study of extra-smooth functions.

I hope this helps.

EDIT You may hate to hear this, but integral calculus is a lot harder than differential calculus, in large part because it uses differential calculus intensively. You may want to repeat this course or maybe take an online course if you feel shaky.

 

[soroban]

Hello, StintedVisions!

I learned that if you take the derivative of the function of an object's position,
you find the velocity of that object.

Furthermore, if you set the function of the velocity to zero,
you can find the time that the object reaches its maximum height.

But if the velocity is zero, couldn't that also be the case before the object
is hurled into the air or after the object hits the ground? . No

Traditionally, we do not consider the object before its release
. . nor after it strikes the ground.


Example: We thrown a ball upward from the ground at 96 ft/sec.

We have: .\(\displaystyle \begin{Bmatrix}h(t) &=& 96t - 16t^2 \\ v(t) &=& 96 - 32t \end{Bmatrix}\)


When \(\displaystyle t = 0\!:\;h(0) = 0\text{ ft},\;v(0) = +96 \text{ ft/sec (upward)}\)


If \(\displaystyle v(t) = 0\!:\;96-32t \:=\:0 \quad\Rightarrow\quad t \,=\,3\text{ sec}\)

Then: .\(\displaystyle h(3) \:=\:96(3) - 16(3^2) \:=\:144\text{ ft (max height)}\)


The ball rose for 3 seconds; it will also fall for 3 seconds.
It will strike the ground when \(\displaystyle t = 6.\)

. . \(\displaystyle v(6) \:=\:96 - 32(6) \:=\:-96\text{ ft/sec (downward)}\)

. . \(\displaystyle h(0) \:=\:96(6) - 16(6^2) \:=\;0\text{ ft (of course!)}\)