### Video Transcript

A particle has a velocity given by
𝑣 as a function of 𝑡 equals 5.0𝑡𝑖 plus 𝑡 squared 𝑗 minus 2.0𝑡 cubed 𝑘 meters
per second. What is the particle’s acceleration
vector at 𝑡 equals 2.0 seconds? What is the magnitude of the
particle’s acceleration at 𝑡 equals 2.0 seconds?

Since we’re asked to solve for
particle acceleration at a particular time 2.0 seconds, we know that this is an
instantaneous acceleration. We can write down that given time
value 2.0 seconds as well as the particle’s velocity function 𝑣 of 𝑡. We’ll use this information in part
one to solve for instantaneous acceleration and in part two to solve for the
magnitude of that instantaneous acceleration.

We can begin solving for the
instantaneous acceleration by recalling the mathematical equation that explains that
term. The instantaneous acceleration an
object undergoes is equal to its change in velocity divided by its change in time,
specifically the time derivative of its velocity as a function of time. We can write that our acceleration
as a function of time is equal to the time derivative of velocity.

When we plug in for our velocity
equation and take this time derivative, we find it’s equal to 5.0𝑖 plus 2𝑡𝑗 minus
6.0𝑡 squared 𝑘 meters per second squared. This is our generalized solution
for acceleration. But we want to solve for
acceleration at a particular instant in time, when 𝑡 equals 2.0 seconds. To solve for it, we’ll insert that
time value everywhere that 𝑡 appears in our general acceleration equation. When we calculate this value, we
find it’s equal to 5.0𝑖 plus 4.0𝑗 minus 24𝑘 meters per second squared. That’s the acceleration of our
object when 𝑡 equals 2.0 seconds.

Now that we know the particle’s
acceleration at that time, we wanna solve for the magnitude of that
acceleration. That magnitude, which tells us how
much the particle’s velocity is changing at the instant in time 2.0 seconds, is
equal to the square root of the acceleration’s 𝑥-component squared plus its
𝑦-component squared plus its 𝑧-component squared.

When we look at our instantaneous
acceleration expression for these components, we find the 𝑥-component is 5.0, the
𝑦-component is 4.0, and the 𝑧-component is negative 24. When we enter this expression on
our calculator, we find that, to three significant figures, it’s 24.8 meters per
second squared. That’s the magnitude of the
particle’s acceleration when 𝑡 equals 2.0 seconds.