The brake mechanism used to reduce recoil

[mech-eng]

Homework Statement

The brake mechanism used to reduce recoil in certain types of guns consists essentially of a piston which is attached to the barrel and may move in a fixed cylinder filled with oil as the barrel recoils with an initial velocity v0, the piston moves and oils is forced through orifices in the piston, causing the piston and the barrel to decelerate at a rate proportional to their velocity; that is, $a=-kv$. Express v in terms of t, x in terms of t, v in terms of x. Draw the corresponding motion curves.

Homework Equations

$\ \displaystyle a=\frac{dv}{dt} \$
$\ \displaystyle v=\frac{ds}{dt}$

The Attempt at a Solution

[/B]
Substituting -kv for a in the fundamental formula defining acceleration, a=dv/dt, we write
$\ \displaystyle -kv=\frac{dv}{dt} \$ ; $\ \displaystyle \frac {dv}{dt}=-kdt$

$\ \displaystyle \ int_v_0^v \, dx = \left. -k \int_0^t \, dt \right \$

$\ \displaystyle ln\fracv_{v_0}=-kt$

I have made mistakes when doing latex. So I paste official solution. I understand most of it but the confusing part is that when integrating why the author did not use integration constant?

Source: Vector Mechanics for Engineers by Beer/Johnston.

Thank you.

 

[BvU]

why the author did not use integration constant

He did use one: $v_0$ and the other, $t_0$ is zero (the moment of firing)

Another way to look at this: if F is a primitive of f, then $\int_a^b f = F(b) - F(a)$ (no integration constants: they cancel)

\ \displaystyle \ int_v_0^v \, dx = \left. -k \int_0^t \, dt \right \

\ \displaystyle \int_{v_0}^v \, dx = \left. -k \int_0^t \, dt \right \
$$ \ \displaystyle \int_{v_0}^v \, dx = -k \int_0^t \, dt $$

\fracv --> \frac v

 

[mech-eng]

Another way to look at this: if F is a primitive of f, then ##\int_a^b f = F(b) - F(a) (no integration constants: they cancel)

I have forgotten some rules of integration. Isn't there any integration constant in definite integral?

Thank you.

 

[mech-eng]

\ \displaystyle \int_{v_0}^v \, dx = \left. -k \int_0^t \, dt \right \

But there are left and right in this example. This is confusing for me.

Thank you.

 

[BvU]

Ah, the \right is a vertical line $\ \$ | $\ \$ (I think it's called a strut) , not a backslash. The \left . is then a kind of dummy.

And this $\TeX$ loses track when two underscores are parsed, so the { } are needed -- don't know if that's universal.

$$\ \displaystyle \int_{v_0}^v \, dx = \left. -k \int_0^t \, dt \right |_1^3 $$

Isn't there any integration constant in definite integral

$$F'(x) = f(x) \Rightarrow \int^x f(u) \, du = F(x) + C \ \ {\rm and }\ \left . \int^b_a f(x) \, dx = F \right |^b_a = F(b)-F(a) $$

 

[mech-eng]

$$F'(x) = f(x) \Rightarrow \int^x f(u) \, du = F(x) + C \ \ {\rm and }\ \left . \int^b_a f(x) \, dx = F \right |^b_a = F(b)-F(a) $$

Because definite integral would be a number which is the diffference of two values of a function and these two values have the same constant, these constants will cancel out, right?

Thank you.

 

[mech-eng]

\ \displaystyle \int_{v_0}^v \, dx = \left. -k \int_0^t \, dt \right \
$$ \ \displaystyle \int_{v_0}^v \, dx = -k \int_0^t \, dt $$

Yes, it seems that the last backslash in \ \ displaystyle is overwritten. So why do some people use it?

Thank you.

 

[BvU]

No: $TeX$ sees a right without an argument such as ], ) or |.
The backslash is not an argument but a sign for $TeX$ to parse the next character(s) as keywords

 

[mech-eng]

Ah, the \right is a vertical line \ \ | \ \ (I think it's called a strut) , not a backslash. The \left . is then a kind of dummy

What does dummy mean you have used above?

https://www.ahdictionary.com/word/search.html?q=dummy

Thank you.

 

[BvU]

\left and \right are grouping characters for $TeX$. It needs to know where a group begins and ends (e.g. for vertical sizing), so it can't have unpaired instances. But if you want a single delimiter you can pair with \left . or \right . : the dot gives you an invisible delimiter.
(it doesn't do anything, it acts as a dummy, a placeholder).

 

[mech-eng]

\left and \right are grouping characters for $TeX$. It needs to know where a group begins and ends (e.g. for vertical sizing), so it can't have unpaired instances. But if you want a single delimiter you can pair with \left . or \right . : the dot gives you an invisible delimiter.
(it doesn't do anything, it acts as a dummy, a placeholder).

I am too amateurish in this topic that when you mentioned TeX I am quite puzzled. Would you explain what the relation between TeX and LaTeX?

Thank you.