WEBVTT
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This is College Physics Answers
with Shaun Dychko.
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A ball is dropped from a height of 1.5 meters
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and it achieves some maximum velocity
which we'll call *v 1*—
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just before it hits the ground here—
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and we'll figure out what that
velocity is in part (a).
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And then it rebounds
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with a velocity which we'll call *v 1 prime*
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keeping the subscript 1 because it's
a velocity at the same position here
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but this velocity is different because
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the bounce is not perfectly efficient
and so some energy is lost
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and so this rebound velocity, *v 1 prime*,
is going to be less than *v 1* was
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when it was going downwards.
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And then it will reach some maximum
height of 1.10 meters
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which we'll call position 2 and it will have
some velocity 2
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of zero there as everything does when
it reaches a maximum height,
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velocity is always zero at the very maximum
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of an arc or of something just
going through the air.
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So we find *v 1* going downwards
here in part (a);
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in part (b), we find *v 1 prime*,
the rebound velocity;
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in part (c), we find the acceleration
of this tennis ball
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when it's in contact with the ground
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and then in part (d), we figure out how much
the tennis ball is compressed.
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So it starts with some spherical
diameter here
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and gets compressed into this ellipse
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and we'll figure out how much
that compression is.
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Okay.
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So we are going to figure out
what is velocity *v 1* then
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and we know that *v 1 squared* equals
*v initial squared*
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minus 2 times acceleration due to gravity
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multiplied by the final height
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minus the initial height—that's equation 77—
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and we are gonna solve for *v 1* by taking
the square root of both sides
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and that is the plus or negative square root
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and we have to use our knowledge
of the physical situation
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to decide whether it should be
positive or negative.
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Now because the tennis ball is moving down
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and we usually take upwards to be
the positive direction
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that means this velocity is
going to be negative.
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So we choose the negative here
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and we have the negative square root
of 0 meters per second squared—
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that's *v naught* is zero, it was dropped
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and so whenever the question says
the word 'dropped' that means
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we are given no initial velocity—
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minus 2 times acceleration due to gravity,
9.80 meters per second squared,
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multiplied by the final height which is
0 meters at the ground level, *y 1*,
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minus the initial height of 1.50 meters
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and this works out to negative
5.42 meters per second.
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In part (b),
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we want to know the rebound
velocity, *v 1 prime*,
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and we know that *v 2 squared*, the velocity
at the very top of its rebound
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when it's at a height of 1.10 meters,
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*v 2 squared* equals the initial velocity,
*v 1 prime squared*,
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minus 2*g* times the final height of *y 2*
minus the initial height of *y 1*
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so it gets to this final height, *y 2*,
and it started down here at *y 1*.
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And we are gonna solve for *v 1 prime* by
adding 2*g y 2* minus *y 1* to both sides
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and this *v 2* is 0 so we are just gonna
make that term disappear
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and switching the sides around,
we have *v 1 prime*
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on the left here and we have also taken
the square root of both sides
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and so that's why we have a square root
of 2*g* times *y 2* minus *y 1* here.
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And we'll take the positive now
because we can see that
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the rebound is going upwards
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so *v 1 prime* is positive.
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So we have positive square root of 2 times
acceleration due to gravity
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multiplied by the final height
of 1.10 meters
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minus the initial height at the ground
of 0 meters
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giving us a velocity, *v 1 prime*, of
4.64 meters per second.
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The acceleration of the tennis ball,
while it's in contact with the ground
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is the final velocity, which is
the rebound velocity,
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minus the initial downward velocity just
before it contacted the ground
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divided by the time it spends
in contact with the ground
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which we are told is 3.5 milliseconds.
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So we have 4.6433 meters per second—
rebound velocity—
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minus the negative—
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and it's important to notice that there's
a minus from the formula
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and then a negative for the velocity *v 1*
together is gonna make a positive—
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5.4222 meters per second divided by
3.50 milliseconds written as times
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10 to the minus 3 seconds giving us
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2.88 times 10 to the 3 meters
per second squared
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and that's a positive acceleration.
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Okay.
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So in part (d),
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we are going to figure out how much
the tennis ball is compressed
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while it's in contact with the ground.
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So we consider this initial moment when
it just begins to contact the ground
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and at this point,
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we are gonna consider this very top
point of the ball
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and we'll follow it through to the point
where the ball is fully compressed
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and at this point,
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when the ball is compressed, this point at
the top of the ball will have zero velocity
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whereas initially, it has a velocity of
negative 5.4222 meters per second
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and I'm calling this *v naught*,
which is sort of a relabelling
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of what I called *v 1* before
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but let's reset our subscripts here
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and call this *v naught* so that we can
write our formulas
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the way we are used to seeing them.
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Okay.
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So this point has a velocity
*v naught* downwards
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and it has an acceleration upwards
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due to being in contact with the ground
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and that acceleration is what
we calculated in part (c).
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And then despite the acceleration upwards,
it is nevertheless still going down because
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the acceleration has not been
applied for long enough to
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reverse the direction of this point
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and so it's continuing down despite
an acceleration upwards
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but it's slowing down very rapidly
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to the point where it finally gets
to a velocity of zero
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and we want to know what is
the difference
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between these two positions of
the top of the ball;
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that will be our compression,
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that is this initial height of *y naught* minus
the final height of *y*.
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So we have equation 54 which tells us
that the final velocity squared equals
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the initial velocity squared plus
2 times acceleration
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times the final position minus
the initial position
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and we are gonna solve for
*y* minus *y naught*
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and we'll do that by subtracting *v naught*
squared from both sides
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and then also noticing this term is zero
because the final velocity is zero
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and we have, after switching the sides around,
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we have 2*a* times *y* minus *y naught*
equals negative *v naught* squared.
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Then we are gonna divide both sides by
negative 2*a*
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and so these negative make a positive so
that's positive *v naught* squared over 2*a*
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and on the left side,
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this negative is gonna switch the signs
of these two terms around
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so that we have *y naught* minus *y*
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and that's a bit nicer to write it
this way because
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we know that that's going to give us
a positive number
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and it's just nice to express our compression
as a positive amount, a magnitude.
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So we have *y naught* minus *y* equals
*v naught* squared over 2*a*
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so that's negative 5.4222 meters per
second squared divided by
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2 times the acceleration that
we calculated in part (c)
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and that gives 0.0051116 meters
which is 5.11 millimeters.
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So it compresses about half a centimeter
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while it's in contact with the ground.