### Video Transcript

A body was projected at 16 meters
per second up a smooth plane inclined at an angle ๐ผ to the horizontal, where sin ๐ผ
is 45 over 49. Determine the time that the body
took to return to the point of projection. Take ๐ to be equal to 9.8 meters
per square second.

To answer a question like this,
weโre going to begin by simply sketching a diagram. We have a smooth plane inclined at
an angle ๐ผ to the horizontal. Now, the fact that the plane is
smooth indicates to us that there will be no frictional forces acting on the
body. Weโre also told that sin of ๐ผ is
equal to 45 over 49. Now, what weโre not going to do is
use this equation to work out the value of ๐ผ. Instead, weโre going to draw a
little right-angle triangle with an included angle of ๐ผ. We know that sin of ๐ผ is opposite
over hypotenuse, so the opposite side in this triangle must be 45 units and the
hypotenuse must be 49 units.

Then, we can use the Pythagorean
theorem to find the length of the missing side. This theorem says that the sum of
the squares of the two shorter sides is equal to the square of the hypotenuse. So, if we let the side weโre
looking for be equal to ๐ units, we get ๐ squared plus 45 squared equals 49
squared. Thatโs ๐ squared plus 2025 equals
2401. Then, we subtract 2025 from both
sides, so ๐ squared is 376, meaning ๐ is the square root of this value. Thatโs two times root 94. Now, thatโs really useful because
we now know the length of the adjacent side. And this will allow us to calculate
exact values for cos ๐ผ and, in fact, for tan ๐ผ, should we need that too.

But letโs go back to our
diagram. The body is being projected up a
smooth plane. Now, what this means is the body
exerts a downward force on the plane due to its mass. The downward force is mass times
gravity. Letโs call that ๐๐ since weโre
not actually given the mass of the body. We are told that ๐ is equal to 9.8
meters per square second. But weโll leave it as ๐ for
now. Thereโs only one more force that
weโre interested in. And thatโs the normal reaction
force of the plane on the body. Remember, that acts perpendicular
to the plane. So, what next? Well, weโre given an initial
velocity. We might call that ๐ข or ๐ฃ sub
zero. And thatโs equal to 16 meters per
second.

Weโre trying to find the amount of
time that it took for the body to return to the point of projection. So what weโre going to do is
actually begin by finding the amount of time it takes before the body comes to rest,
in other words, when its final velocity ๐ฃ is equal to zero. And in order to do this, we need to
calculate the acceleration. And so, weโll use the formula ๐น
equals ๐๐. Force is mass times
acceleration. Weโre going to carry this
calculation out parallel to the plane. And so, we add a right-angle
triangle to our weight force. And we need to add this triangle
because this force acts vertically downwards, not parallel, not perpendicular to the
plane. The component of the weight that
acts parallel to the plane is the side Iโve called ๐ฅ. And we, of course, have this
included angle. This angle is ๐ผ.

We go back to the convention for
labeling our triangles, and we see weโre looking to find the opposite. And we know the hypotenuse or at
least we have an expression for the hypotenuse in terms of ๐. sin ๐ผ is opposite
over hypotenuse, so sin ๐ผ is ๐ฅ over ๐๐. By multiplying both sides of this
equation by ๐๐, we get ๐ฅ is equal to ๐๐ sin ๐ผ. sin ๐ผ is 45 over 49, so we get
๐ฅ equals 45 over 49 ๐๐. Now, the particleโs moving up the
plane, so letโs define a positive direction as being up the plane. The force is acting in the opposite
direction, so the resultant force that acts parallel to the plane is negative 45
over 49 ๐๐. Thatโs equal to mass times
acceleration, ๐๐.

And next, we notice that we can
divide through by ๐. Now, remember, we can only divide
through by a variable if weโre certain itโs not equal to zero. This represents a mass, so it
absolutely canโt be. And weโve now found the
acceleration of the body. Itโs negative 45 over 49 ๐ meters
per square second. Now, it actually makes a lot of
sense that the acceleration is negative. There are no forces helping to move
the body up the plane, so we would assume itโs slowing down. Now, we didnโt actually need to
calculate the value of cos ๐ผ. And so, we can get rid of our
right-angle triangle. Itโs always sensible to construct
these when given values for sin ๐ผ, cos ๐ผ, or tan ๐ผ because it can create easier
calculations later on down the line.

Now, we said weโre going to begin
by calculating the amount of time it takes for the particle to reach a speed of
zero. So, its initial speed ๐ข was 16
meters per second, and its final speed ๐ฃ is zero. Its acceleration is negative 45
over 49 ๐. And weโre looking for ๐ก. And so, weโre going to now use one
of the equations of constant acceleration. The one that weโre looking for that
links ๐ข, ๐ฃ, ๐, and ๐ก is ๐ฃ equals ๐ข plus ๐๐ก. We replace the values we know in
this formula, and we get the equation zero equals 16 minus 45 over 49 ๐๐ก. We solve first by adding 45 over 49
๐๐ก to both sides and then dividing through by 45 over 49 ๐ or 45 over 49 times
9.8.

And so, in doing so, we see that
the time it takes for the body to reach a speed of zero is 16 over nine seconds. So, how does this help us calculate
the total time that the body takes to return to its starting point? Well, in a mathematically perfect
world โ that is, one where, in this case, thereโs no friction nor air resistance or
anything like that โ the time it takes to reach that point where ๐ฃ equals zero will
be the exact same time it takes to return to the starting point. And so, letโs define the total time
as being capital ๐. And we know that weโre just going
to double 16 over nine. 16 over nine times two is 32 over
nine, so we can say that the total time the body takes to return to the point of
projection is 32 over nine seconds.