The position of a particle is given
by position as a function of time equals 4.0𝑡 squared in the 𝐢 direction minus 3.0
in the 𝐣 direction plus 2.0𝑡 cubed in the 𝐤 direction meters. What is the velocity of the
particle at 0.0 seconds? What is the velocity of the
particle at 1.0 seconds? What is the average velocity of the
particle between 0.0 seconds and 1.0 seconds?
Let’s call the velocity of the
particle at 0.0 seconds 𝑣 of 0.0 s and the velocity of the particle at 1.0 seconds
𝑣 of 1.0 s. And we’ll call the average velocity
of the particle between 0.0 and 1.0 seconds 𝑣 sub avg.
Solving for these values will
involve taking a time derivative of the position of the particle function. So let’s begin on that now.
We can recall that the velocity of
a particle 𝑣 is equal to the time derivative of the position of the particle. Applying this relationship to our
scenario, the velocity of the particle as a function of time is equal to the time
derivative of 𝐫 of 𝑡.
We take that derivative term by
term. The derivative of the 𝐢th
component of 𝐫 is 8.0 times 𝑡; the derivative of the 𝐣th component of 𝐫 is zero;
and the derivative of the 𝐤th component of 𝐫 is 6.0 times 𝑡 squared.
And because we’ve differentiated
with respect to time, instead of units of meters we now have units of meters per
second. We can now move ahead with solving
for the velocity at 0.0 seconds.
When we plug in zero for 𝑡, all
components of velocity go to zero, so the velocity at 0.0 seconds equals zero meters
Now we can move onto solving for
velocity when time equals 1.0 seconds. When we plug in 1.0 seconds for 𝑡,
then our velocity equals 8.0𝐢 plus 6.0𝐤 meters per second.
Now we’re ready to move onto
solving for 𝑣 sub avg. The particle’s average speed
between 0.0 seconds and 1.0 seconds, that average speed is equal to the position of
the particle at 1.0 seconds minus the position of the particle at 0.0 seconds
divided by change in time which works out to 1.0 seconds.
If we plug in 1.0 seconds for 𝑡 in
the 𝐫 of 𝑡 equation, then we find that’s equal to 4.0𝐢 minus 3.0𝐣 plus
Then when we evaluate that function
at time 𝑡 equals 0.0 seconds, it equals simply minus 3.0𝐣.
Since we’re subtracting that term
in the first place, it becomes positive 3.0𝐣. Looking at the numerator of this
fraction, we see that the 𝐣 component cancels out so that our final answer for the
average speed of the particle over this time interval is 4.0𝐢 plus 2.0𝐤 meters per
This is the particle’s average
velocity for the first second of its motion.