A car of mass 1350 kilograms moves
in a straight line such that, at time 𝑡 seconds, its displacement from a fixed
point on the line is given by 𝑠 equals six 𝑡 squared minus three 𝑡 plus four
meters. Find the magnitude of the car’s
momentum at 𝑡 equals three seconds.
We’re looking for the magnitude of
the car’s momentum. Momentum is a course of vector
quantity. But we’re 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. It’s 1350 kilograms.
But we’re not explicitly told the
velocity of the car anywhere in the question. What we are told is its
displacement 𝑠 equals six 𝑡 squared minus three 𝑡 plus four meters, where 𝑡 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 d𝑡 of the displacement at that point in time.
So we can call the velocity 𝑣 and
displacement 𝑠 as it is in the question. And we can substitute the
expression we have for 𝑠 in terms of 𝑡. We’re told that the displacement 𝑠
is six 𝑡 squared minus three 𝑡 plus four. And we can differentiate term by
The derivative of 𝑡 squared with
respect to 𝑡 is two 𝑡. And so the derivative of six 𝑡
squared is six times two 𝑡, which is 12𝑡. The derivative of 𝑡 with respect
to 𝑡 is just one. And so the derivative of three 𝑡
with respect to 𝑡 is three. And the derivative of a constant is
zero. And so the constant term four
doesn’t contribute anything to the velocity.
We found therefore that the
velocity at time 𝑡 is 12𝑡 minus three. And as the displacement was
measured in metres and the time in seconds, this velocity has units of metres per
So now that we have the velocity at
time 𝑡, we can substitute it in to our equation for momentum. The momentum is 1350 times 12𝑡
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
𝑡. But we’re only interested when 𝑡
is three seconds.
So we substitute 𝑡 equals three
here. 𝑡 is three and so 12𝑡 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. That’s 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, it’s also the magnitude of
the car’s momentum.
This is therefore the answer to our
question, 44550 kilogram metres per second.