### Video Transcript

A particle is moving in a straight
line such that its acceleration at time π‘ seconds is given by π equals two π‘
minus 18 meters per seconds squared, where π‘ is greater than or equal to zero. Given that its initial velocity is
20 meters per second and its initial displacement is zero meters, find an expression
for the displacement of the particle at time π‘.

Remember, acceleration is the rate
of change of the velocity of an object. This means we differentiate the
function for velocity to find the function for acceleration. We can conversely say that the
integral of the function for acceleration will provide us with the function for
velocity. Similarly, velocity is equal to the
derivative of displacement with respect to time. So we can say that to find
displacement, we can integrate the function for velocity with respect to time.

To answer this question then, weβre
going to need to integrate our function for acceleration with respect to time
twice. Throughout this process, weβll use
the fact that we know its initial velocity and its initial displacement. This will help us to find a
particular expression for displacement. So velocity is the indefinite
integral of two π‘ minus 18 with respect to π‘. The integral of two π‘ is two π‘
squared over two. The integral of negative 18 is
negative 18π‘. And of course, we mustnβt forget
that we have this constant of integration π. And we see that π£ is equal to π‘
squared minus 18π‘ plus π. This is known as the general
expression. We, however, know that its initial
velocity is 20 meters per second. So we can say that when π‘ is equal
to zero, π£ is equal to 20.

And we can use this information to
find the particular expression for velocity. We substitute π‘ equals zero and π£
equals 20 into this equation. And we obtain 20 equals zero squared
minus 18 times zero plus π, which gives us 20 equals π. And we have the expression for
velocity at time π‘. Itβs π‘ squared minus 18π‘ plus
20. Weβre gonna integrate one more time
to find the expression for displacement. Itβs the integral of π‘ squared
minus 18π‘ plus 20 with respect to π‘.

This time, the integral of π‘
squared is π‘ cubed over three. The integral of negative 18π‘ is
negative 18π‘ squared over two. The integral of 20 is 20π‘ and we
have a constant of integration. Notice Iβve called this π instead
of π because weβve already used π in this question. So π is equal to π‘ cubed over
three minus nine π‘ squared plus 20π‘ plus π. Once again, we have enough
information to work out the value of π. We know the initial displacement is
zero meters. So when π‘ is equal to zero, π is
equal to zero. And we substitute these values
in. And we obtain zero equals zero
cubed over three minus nine times zero squared plus 20 times zero plus π, which
tells us that zero is equal to π. And weβre done! We found the expression for the
displacement of the particle at time π‘. Itβs π‘ cubed over three minus nine
π‘ squared plus 20π‘ meters.