Intuition of Impulse Formula $J = \sum F \Delta t$

Question

I understand that

$$\begin{align} J = \sum F \Delta t &= \Delta p \\ \sum F &= \frac{\Delta p}{\Delta t} \\ &= \frac{mv_2 - mv_1}{\Delta t} \\ &= m \cdot \frac{\Delta v}{\Delta t} \\ &= ma \end{align}$$

$J$ is impulse, $p$ is momentum

However, by just looking at the equation

$$J = \sum F \Delta t$$

I seem to think that impluse is high if for same force, time is higher. But I think that's wrong? I think for a certain force, if I exert it in a short time theres a larger impluse? How can I then understand the above equation more intuitively without expanding it out?

Answer 1

"I seem to think that impluse is high if for same force, time is higher."

This is correct, to develop an intuitive understanding, we must just realize that impulse simply refers to a change in momentum. So lets say you apply a force of 1N to an object. Is the object going to have a larger change in momentum if you apply 1N for 1s or 1N for 5s?

It should seem clear then that a longer exertion time leads to a larger impulse.