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We would now like to use
Newton's second law to relate

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impulse to change in momentum.

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So again, let's
look at our set up.

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We have an object
m, a velocity V,

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and, let's say, here the
picture as t initial.

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And we have some
initial velocity.

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And then a little bit later
in time, we have time t final.

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The moment the
velocity has changed.

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And that's because throughout
this time interval,

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we're applying an impulse.

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We can call this
the i hat direction.

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Now recall that our
definition of impulse,

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it's a vector quantity.

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It's equal to the
integral of the force.

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Now when I write force
of t, I mean force

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as a function of time.

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And that's our dummy
variable t prime.

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It's not force times time, but
force is a function of time.

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And we're integrating
from the initial

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to the final time period.

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Now here's where
we use the second--

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the version of
Newton's second law,

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which is that force causes
the momentum of an object

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to change.

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So we can write this integral t
prime t initial, t prime equals

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t final of dp dt.

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I'll just make a
note that we've now

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applied Newton's second
law, and because we're using

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our dummy variable dt prime

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And you can see that the
two dt primes cancel.

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And this just
becomes the integral

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from t initial t prime
to t final of dp.

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And this is a pure differential.

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And so we end up with impulse
is the momentum at t final minus

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the momentum at t initial.

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Now recall, this is
a vector integral.

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But a vector integral is
just three separate intervals

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for each component.

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So each component of impulse
satisfies this equation.

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For instance, the x
component of impulse

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is how the x component of
momentum is changing in time.

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Now generally, when we
write a final momentum

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in our final state minus the
momentum during initial state,

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we can call this the
change in momentum.

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So down here, we would have
change in the x direction.

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And so in conclusion, we
now have an integral way

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of casting Newton's second law.

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We have that impulse
causes momentum to change.

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And so we can see that
the si units of impulse

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are the same as the
si units of momentum,

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which we saw before was
kilogram meter inverse seconds.