### Video Transcript

A body moves along the π₯-axis. Such that, at time π‘ seconds, its
displacement from the origin is given by π is equal to six π‘ to the fourth
power minus π‘ cubed minus three π‘
squared minus four π‘ plus three metres. For values of π‘ where π‘ is
greater than or equal to zero. What is its velocity when its
acceleration is equal to zero?

The question tells us that the body
is moving along the π₯-axis. And it gives us a function for its
displacement from the origin at a time of π‘ seconds, which is a polynomial, in
terms of metres. The question wants us to use this
information to find the velocity of the particle when its acceleration is equal to
zero.

We can start by asking the
question, when will the body have an acceleration which is equal to zero? Weβre given an expression to
calculate the displacement from the origin of the body after π‘ seconds. And since the body is moving in a
straight line, we know that, for a displacement π , the rate of change of
displacement gives us the velocity. And the rate of change in velocity
gives us the acceleration.

So to find the times when the body
has an acceleration of zero, we need to find the second derivative of our
displacement function with respect to π‘. Weβll start by finding the first
derivative of π with respect to π‘, which is of course also equal to the velocity
of the body. This gives us the derivative of six
π‘ to the fourth power minus π‘ cubed minus three π‘ squared minus four π‘ plus
three with respect to π‘.

Since weβre differentiating a
polynomial, we can differentiate this using the power rule for differentiation. We multiply by the exponent and
then reduce the exponent by one. This gives us the velocity of the
body at the time π‘ is 24π‘ cubed minus three π‘ squared minus six π‘ minus
four.

We can now use this to find the
acceleration of the body at the time π‘. Itβs the second derivative of the
displacement with respect to π‘, which is of course just equal to the rate of change
of the velocity with respect to time. So the acceleration is equal to the
derivative of 24π‘ cubed minus three π‘ squared minus six π‘ minus four with respect
to π‘.

We can differentiate this using the
power rule for differentiation. And this gives us 72π‘ squared
minus six π‘ minus six. And we wanted to find when the
acceleration was equal to zero. So we set our acceleration equal to
zero. This means that zero is equal to
72π‘ squared minus six π‘ minus six. And this is a quadratic
equation. So we can solve for the values of
π‘.

We can simplify the equation by
dividing both sides of the equation by six. This gives us that zero is equal to
12π‘ squared minus π‘ minus one. Thereβs a few different ways we
could solve this quadratic equation. For example, we could use the
quadratic formula. However, in this case, we can
factor it fully.

We want our coefficients to π‘ to
multiply to give us 12. For example, we could try four and
three. Then if we want our constants to
multiply to give us negative one. We can see that four π‘ plus one
multiplied by three π‘ minus one gives us 12π‘ squared minus π‘ minus one. Then this will be equal to zero
when one of the factors is equal to zero. So we either have that π‘ is equal
to negative a quarter or π‘ is equal to one-third.

However, in this case, π‘
represents the time. In fact, weβre told that π‘ must be
greater than or equal to zero. So in fact, we only have one
solution when π‘ is equal to one-third. So since the question wants us to
find the velocity of the body when the acceleration is equal to zero. And we just showed the only time
the acceleration is equal to zero is when π‘ is equal to one-third. We need to find π£ evaluated at
one-third.

Substituting π‘ is equal to
one-third into the velocity gives us 24 times one-third cubed minus three times
one-third squared minus six times one-third minus four. Which we can evaluate to give us
negative 49 divided by nine. In fact, since weβre measuring the
time in seconds and the displacement in metres, our velocity, which is the rate of
change of displacement, must have the units metres per second.

Therefore, weβve shown if a body
moves along the π₯-axis such that, at a time of π‘ seconds, its displacement from
the origin is given by π is equal to six π‘ to the fourth power. Minus π‘ cubed minus three π‘
squared minus four π‘ plus three metres, where π‘ is greater than or equal to
zero. Then its velocity when its
acceleration is equal to zero is negative 49 divided by nine metres per second.