Impulse integration for a Tennis Racket hitting a Tennis Ball

AI Thread Summary
Integrating impulse over time from t_i to t_f is necessary because impulse is a function of time, represented as J(t). This approach allows for the calculation of cumulative impulse during a specific time interval. Using J directly without specifying time bounds can lead to confusion, as impulse must be evaluated over time to be meaningful. The discussion clarifies that if impulse values at endpoints are known, they can be used, but otherwise, time integration is essential. Understanding J as a time-dependent function resolves the initial confusion regarding the integration limits.
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Homework Statement
Please see below
Relevant Equations
Please see below
For this,
1684125887202.png

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!
 
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ChiralSuperfields said:
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
 
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ChiralSuperfields said:
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.
 
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haruspex said:
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!
 
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