### Video Transcript

A particle is projected vertically
upward at seven meters per second from a point 38.7 meters above the ground. Find the maximum height the
particle can reach. Consider the acceleration due to
gravity to be 9.8 meters per second squared.

Alright, so if we say that this is
ground level, then if we go 38.7 meters up from there, we have our particle where it
is projected with an upward velocity of seven meters per second. This tells us that our particle
will continue moving upward. But we know that under the
influence of gravity, it will slow down more and more until eventually it comes to
rest. At this point, it has reached its
maximum height. We’ll call it ℎ sub max. This is the height we want to solve
for, and we can see it’s equal to 38.7 meters plus this height here. We’ll call this height ℎ. And since as our particle moves
across this height it’s accelerating uniformly under the influence of gravity, we
can use an equation of motion to solve for ℎ. Specifically, we’ll use the
relationship that an object’s final velocity squared is equal to its initial
velocity squared plus two times its acceleration times its displacement.

Applying this relationship to our
situation, we’ll say 𝑣 sub 𝑓 squared equals 𝑣 sub 𝑖 squared plus two times 𝑔,
that’s our particle’s acceleration, multiplied by ℎ. Going back over to our sketch, if
we say that our particle’s position here is its initial position, where it had an
initial velocity 𝑣 sub 𝑖, and our particle’s position here at its maximum height
is its final position, then we can say that its final velocity 𝑣 sub 𝑓 is zero,
meaning that this is then the equation we have to solve for the height ℎ.

At this point, we can set up a sign
convention where we say that motion upward is positive and motion downward is
therefore negative. This is useful to us because the
acceleration due to gravity is downward, while our initial velocity, what we’ve
called 𝑣 sub 𝑖, is upward. And so leaving off the units, this
means that we would use a value of positive seven for 𝑣 sub 𝑖 and negative 9.8 for
𝑔. Substituting in these values, if we
then subtract seven squared from both sides of this equation, we have that negative
seven squared is equal to two times negative 9.8 times ℎ. And both negative signs drop
out. And if we then divide both sides of
the equation by two times 9.8, we find that ℎ is equal to 49, that’s seven squared,
divided by two times 9.8. That’s equal to 2.5. And we’ll include the units of
meters.

Recall though that this isn’t our
answer because ℎ is just one part of ℎ max. ℎ max is equal to 38.7 meters plus
ℎ or 38.7 meters plus 2.5 meters. And adding these together, we get
41.2 meters. This is the maximum height our
particle can reach.