Impulse integration for a Tennis Racket hitting a Tennis Ball

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

For this,

Can someone please tell me why they integrate the impulse over from $t_i$ to $t_f$? Why not from $j_i$ to $j_f$? It seems strange integrating impulse with respect to time.

Many thanks!

 

[topsquark]

Homework Statement: Please see below
Relevant Equations: Please see below

For this,
View attachment 326569
Can someone please tell me why they integrate the impulse over from $t_i$ to $t_f$? Why not from $j_i$ to $j_f$? It seems strange integrating impulse with respect to time.

Many thanks!

If you are using
$\displaystyle \int d \textbf{J}$
and you have the $\textbf{J}$'s at the endpoints, the use the $\textbf{J}$'s.

If you don't have the $\textbf{J}$'s then you need to use
$\displaystyle \int \textbf{F}(t) \, dt$

-Dan

 

[haruspex]

Can someone please tell me why they integrate the impulse over from $t_i$ to $t_f$? Why not from $j_i$ to $j_f$? It seems strange integrating impulse with respect to time.

If you think of $\vec J$ as the cumulative impulse given over some period of time then $\vec J=\vec J(t)$ and it is reasonable to write $\int_{t=t_i}^{t_f}d\vec J(t)$. But omitting the "t=" from the bounds is a bit naughty.

 

[ChiralSuperfields]

If you think of $\vec J$ as the cumulative impulse given over some period of time then $\vec J=\vec J(t)$ and it is reasonable to write $\int_{t=t_i}^{t_f}d\vec J(t)$. But omitting the "t=" from the bounds is a bit naughty.

Thank you for your reply @topsquark and @haruspex!

@haruspex, now that you say J is a function of t I think that helps.

Many thanks!