### Video Transcript

A body of mass nine kilograms is
released from rest on a smooth inclined plane. It moves a distance of 25.2 meters
in the first four seconds of its motion. If the body is projected upward
along a line of greatest slope on the same plane, with an initial velocity of 12.6
meters per second, how far does it travel before coming to instantaneous rest? Take ๐ to equal 9.8 meters per
second squared.

Alright, so here we have a smooth
inclined plane. And weโre told that a body with a
mass weโll call ๐ of nine kilograms is released from rest on the plane. Since thereโs no frictional force
opposing its motion, the body starts to slide downward. And weโre told that after four
seconds, itโs moved a distance on the plane of 25.2 meters. This information is given to us so
we can solve for the bodyโs acceleration when itโs on the plane. Weโll need to know that to answer
this question of how far the body moves up the plane projected at an initial
velocity before coming to rest.

To solve then for this bodyโs
acceleration as it slides down the plane, itโs important to realize that this
acceleration is constant. If we call the angle of inclination
of our smooth plane ๐, then ๐, the acceleration, is equal to ๐ times the sin of
๐. The point here is that acceleration
is constant because ๐ and ๐ are constant. Therefore, we can calculate the
acceleration of our body using this given information by referring to the equations
of motion. We wonโt list all four of those
equations. Here, weโll just write down
one. This equation of motion says that
the displacement of a body is equal to its initial velocity times the time elapsed
plus one-half its acceleration times that elapsed time squared.

Now, in our case, the distance
traveled is ๐, and the change in time is ฮ๐ก. And because our body is released
from rest, that means that ๐ฃ sub ๐ in this equation, the bodyโs initial velocity,
is zero. Therefore, ๐ equals one-half ๐
times ฮ๐ก squared. If we multiply both sides of this
equation by two and divide both sides by ฮ๐ก squared, we get an equation where ๐ is
the subject. We wonโt yet solve for ๐, but we
will come back to this equation in a moment.

For now, letโs consider this second
stage of motion for our body. Weโre to imagine itโs projected up
the plane at 12.6 meters per second. We know that thanks to the effects
of gravity, the body will slow down over time and eventually it will come to a stop
just instantaneously.

If we call the distance the body
travels as it slows down from 12.6 to zero meters per second capital ๐ท, then we can
see that itโs this distance we want to solve for to answer our question. To do this, weโll once again refer
to the equations of motion. To see what that equation is, letโs
clear away our problem statement, and weโll now refer to this equation of
motion. It says that the final velocity of
an object squared equals its initial velocity squared plus two times its
acceleration times its displacement. Written in terms of our variables,
we could say that ๐ฃ๐ squared equals ๐ฃ๐ squared plus two times ๐ times capital
๐ท.

And notice this: because our object
ends up at rest, that means that ๐ฃ sub ๐ is equal to zero. So we can write that zero equals ๐ฃ
sub ๐ squared plus two ๐๐ท. Since itโs ๐ท we want to solve for,
letโs rearrange this expression. We get that ๐ท equals negative ๐ฃ
sub ๐ squared over two ๐. And we can now substitute in the
expression we have for the acceleration ๐. Now, regarding this minus sign, if
we consider ๐ฃ sub ๐, the initial velocity of our body, to be positive, then that
means weโre saying that motion up the incline is in the positive direction. Therefore, motion the opposite way
is in the negative direction. And thatโs the way our acceleration
๐ will point.

By this sign convention, ๐ is a
negative quantity, and therefore the negative sign in numerator and denominator will
cancel out. Weโre finally ready to plug in for
๐ฃ sub ๐, ๐ท, and ฮ๐ก. Leaving out units, we use a value
of 12.6 for ๐ฃ sub ๐, 25.2 for ๐ท, and four for ฮ๐ก. When we compute this fraction, we
actually get a result of exactly 25.2. This distance is in meters. So we can say that if our body is
projected up the incline at 12.6 meters per second, the distance that will move
before it comes to rest is 25.2 meters.