### Video Transcript

A car of mass 1350 kilograms moves
in a straight line such that, at time ๐ก seconds, its displacement from a fixed
point on the line is given by ๐ equals six ๐ก squared minus three ๐ก plus four
meters. Find the magnitude of the carโs
momentum at ๐ก equals three seconds.

Weโre looking for the magnitude of
the carโs momentum. Momentum is a course of vector
quantity. But weโre only interested in the
magnitude of this quantity. Now you might know that the
momentum of an object is the product of its mass and velocity. And this is good news because we
know the mass of the car. Itโs 1350 kilograms.

But weโre not explicitly told the
velocity of the car anywhere in the question. What we are told is its
displacement ๐ equals six ๐ก squared minus three ๐ก plus four meters, where ๐ก is
the time in seconds. And we can use this displacement to
find the velocity as the velocity at any given point in time is just the
instantaneous rate of change d by d๐ก of the displacement at that point in time.

So we can call the velocity ๐ฃ and
displacement ๐ as it is in the question. And we can substitute the
expression we have for ๐ in terms of ๐ก. Weโre told that the displacement ๐
is six ๐ก squared minus three ๐ก plus four. And we can differentiate term by
term.

The derivative of ๐ก squared with
respect to ๐ก is two ๐ก. And so the derivative of six ๐ก
squared is six times two ๐ก, which is 12๐ก. The derivative of ๐ก with respect
to ๐ก is just one. And so the derivative of three ๐ก
with respect to ๐ก is three. And the derivative of a constant is
zero. And so the constant term four
doesnโt contribute anything to the velocity.

We found therefore that the
velocity at time ๐ก is 12๐ก minus three. And as the displacement was
measured in metres and the time in seconds, this velocity has units of metres per
second.

So now that we have the velocity at
time ๐ก, we can substitute it in to our equation for momentum. The momentum is 1350 times 12๐ก
minus three. And as the mass was measured in
kilograms and the velocity was measured in metres per second, this momentum is
measured in kilograms metres per second. This is the momentum at anytime
๐ก. But weโre only interested when ๐ก
is three seconds.

So we substitute ๐ก equals three
here. ๐ก is three and so 12๐ก is 36. And probably, the most sensible
thing to do is just to put this into our calculator to get a momentum of 44550
kilogram metres per second. Thatโs the momentum of our car
after three seconds. And as this car is moving in a
straight line and the momentum turned out to be positive, itโs also the magnitude of
the carโs momentum.

This is therefore the answer to our
question, 44550 kilogram metres per second.