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# Differential equations with rectilinear motion watch

1. I'm currently studying Mechanical Engineering 1st year, however, I feel this problem falls into Physics/Maths also. The following question is in "Engineering Mechanics, Dynamics, 6th Edition", by "J. L. Mriam and L. G. Kraige", page 29 but here are pictures:

The question:
Spoiler:
Show

The soltuion:
Spoiler:
Show

The question is with regards to the integration of the equation which is next to the number 2. This is ds/(sqrt(v^2-(k^2*s^2))). I do not understand how this then equals 1/k(*(sin^-1(ks/v))).

Appreciate any help.
2. (Original post by No-Idea.)
..
It's integrated using a trigonometric substitution. It's in the form , so we look for a substitution that would cancel this down to a single term. I.e, if you subbed in some form of , and use the identity .

Click the spoiler for the full working. I wouldn't usually do this, but it's a common integral that you'll learn to know by heart, and the maths is slightly tricky.

Spoiler:
Show

Let

---> Taken a factor of out the root, and cancelled it with the . Used the identity

---> Using the initial substitution.

3. (Original post by Phichi)
It's integrated using a trigonometric substitution. It's in the form , so we look for a substitution that would cancel this down to a single term. I.e, if you subbed in some form of , and use the identity .

Click the spoiler for the full working. I wouldn't usually do this, but it's a common integral that you'll learn to know by heart, and the maths is slightly tricky.

Spoiler:
Show

Let

---> Taken a factor of out the root, and cancelled it with the . Used the identity

---> Using the initial substitution.

I see, thanks a lot
4. (Original post by Phichi)
It's integrated using a trigonometric substitution. It's in the form , so we look for a substitution that would cancel this down to a single term. I.e, if you subbed in some form of , and use the identity .

Click the spoiler for the full working. I wouldn't usually do this, but it's a common integral that you'll learn to know by heart, and the maths is slightly tricky.

Spoiler:
Show

Let

---> Taken a factor of out the root, and cancelled it with the . Used the identity

---> Using the initial substitution.

How did you go about getting s=(v/k)*sin(theta)?

Thanks
5. (Original post by No-Idea.)

How did you go about getting s=(v/k)*sin(theta)?

Thanks
Bump..anyone..?
6. (Original post by No-Idea.)

How did you go about getting s=(v/k)*sin(theta)?

Thanks
Ever so sorry, for some reason I've only just seen this!

Like I mentioned in the post, I was looking for a substitution to narrow the expression down to a single term.

We have this:

And to apply

I wanted the root to be in the form:

Where would be either or

I picked to make it , so I could avoid having the negative in the differentiation when switching the variables.

So, we have, and we want .

So we need a substitution for s, where becomes

Thus:

Does that help? Sorry for the long winded answer, you'll learn to do this sort of thing via inspection however, someday.

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