### Video Transcript

A particle’s position varies
according to 𝑥 as a function of 𝑡 equals 3.0𝑡 squared plus 0.50𝑡 cubed
meters. What is the particle’s
instantaneous velocity when 𝑡 equals 2.0 seconds? What is the particle’s average
velocity between 𝑡 equals 1.0 seconds and 𝑡 equals 3.0 seconds?

In part one of this exercise, we
want to solve for instantaneous velocity when 𝑡 equals 2.0 seconds. Then in part two, we’ll solve for
the average velocity of the particle over a time interval centered on that same time
value. This one begins at 1.0 seconds and
ends at 3.0 seconds. In calculating instantaneous
velocity, we can recall that it’s equal to the time derivative of the position as a
function of time of our particle of interest. So in our case, the instantaneous
velocity of the particle as a function of time equals 𝑑 𝑑𝑡, the time derivative
of position as a function of time which we’re given. When we plug in our equation for
position as a function of time and take the time derivative. This derivative results in the
expression 6.0𝑡 plus 1.50𝑡 squared in units of meters per second.

We’ve arrived at a general
expression for instantaneous velocity. But we want to solve for the
velocity at a particular time value, when 𝑡 equals 2.0 seconds. To calculate that value, we plug in
a value of 2.0 seconds for 𝑡 in our expression. When we calculate this value, to
two significant figures, our result is 18 meters per second. That’s the instantaneous velocity
of our particle when 𝑡 equals 2.0 seconds. Next, we move on to solving for the
average velocity of our particle over the time interval 1.0 to 3.0 seconds. This average velocity will be equal
to the position of our particle at 3.0 seconds minus its position at 1.0 seconds
divided by the time interval 3.0 minus 1.0 seconds. Or, in the denominator, 2.0
seconds.

To solve for the position of our
particle at various times, we can use the expression given to us in the problem
statement. When we plug in a value of 3.0
seconds for 𝑡 and calculate the position, we find a result of 40.5 meters. Which we insert for our position
when 𝑡 equals 3.0 seconds. We then do the same thing for 𝑡
equals 1.0 seconds, plugging that value in to our expression and finding a result of
3.50 meters. Which we then insert for our
position when 𝑡 equals 1.0 seconds. We’re now ready to calculate the
average velocity of our particle over the time interval of interest. When we do and round the result to
two significant figures, we find it’s equal to 19 meters per second. So our average velocity and our
instantaneous velocity, even though they’re centered on the same time values, are
not the same.