J-Rock physics from hell: The hide pitching a TV problem.
Legend has it that
hide set a record for the longest distance achieved by a TV set pitched out a
hotel room window. Now Ifve researched this and still have no idea if this
record even exists, but I thought to myselfcHEY!!! THIS WOULD MAKE A YAY BOUNCY
PHYSICS PROBLEM!!!!! So true or not, prepare for suffering at the hands of meeeeeeeeeeee.
In this badly
drawn windows-paint-made diagram, stick-figure-hide is launching Terebi-san (Trans. gMr. TVh) set from a hotel room balcony
(since I was too lazy to drawn a window.). Ifve assumed that his room is about
50 meters up from the ground. hide-sama
pitches terebi-san at a 30 degree angle from the
horizontal, and some seconds later, terebi-san
smashes into the ground 180 m away and goes to TV heaven. Letfs find terebi-sanfs initial velocity (the velocity with which
hide-sama pitched him) and the time it took him to
hit the ground. Neglect air resistance. Ifm too lazy to solve it with air
resistance. lol
The easiest way to
solve this is to divide things up into two components: horizontal and vertical. First, take a close look
at the velocity. Velocity is a vector, so it has direction. This means itfs
going to have happy little components, one vertical component and one
horizontal component:
Can you see how
they sort of form a little triangle?
If you can
remember all your trig, youfll remember that and
(where r is the hypotenuse)
From this you can
derive the little values I have there for voy and vox
So anyway, now
that you have the horizontal and vertical components of the velocity, make your
self a little 2-column table with a ghorizontal stuffh column and a gvertical
stuffh column. List important variables such as time,
displacement, velocity, and acceleration.
Horizontal |
Vertical |
Displacement: x = 180 m Time: t** Initial
velocity: v = v0cos30o
= .866vo m/s Acceleration: 0 m/s2 Final
velocity: No
acceleration, so same as initial velocity. |
Displacement: y = -50 m* Time: t** Initial
velocity: v = v0sin30o
= .5vo m/s Acceleration: -9.81 m/s2 *** Final
velocity: (youfll see we
donft actually have to know this yet.) |
Notes about values:
* Notice I said
gdisplacementh specifically instead of distance. We want the y amount of
distance from where we started. The TV went up for a bit, then down and finally
crashed on the ground 50 below the starting point (although not directly
below), so the final displacement is 50 m below that point, or for
calculations, -50.
** The time is the
same for both horizontal AND vertical travel.
*** The
acceleration acting on Terebi-san is acceleration due
to gravity. Most people take this to be about 9.81. Itfs negative because itfs
pulling DOOOOOOWNward. (duh.)
Real quick, Herefs another bizarre little picture to help you visualize
what exactly is going on with the velocity components and acceleration:
(Even though I
canft draw arrows to save my life, you should sort of be able to tell that the
vertical velocity is changing because of the pull of gravity, while the
horizontal velocity stays the same.
Anyway, now youfll
find your unknowns using kinematic equations.
The calculations
for the horizontal are pretty easy.
Use the equation x = vt.
x = vt
180 = .866vot
This may look
hopeless, but itfs not. Setting up a certain equation for the vertical part
will allow you to make substitutions back and forth.
For the vertical
part, using the equation x = xo
+ vot + ½ at2 gives:
y = yo + t – (9.81/2) t2
(remember that the gxh just refers to displacement. In this
case, itfs displacement in the y (up and down) direction.)
Since youfre using
the hotel room window as the point of origin, and dropping 50 m for your gyh,
you can go ahead and take the initial displacement, yo,
to be 0. Now you have:
-50 = vot – (9.81/2) t2
It looks like we
have two variablescBut wait!!! Wefve got vot
somewhere else too! Go back to the horizontal equation. Solve it for vo and plug into the
vertical equation:
180 = .866vot
vo = 180/.866t = 207.85/t
-50 = vot – (9.81/2) t2
-50 = (207.85/t)t
– (9.81/2) t2
And the tfs in the first term cancel soc
-50 = 207.85 –
(9.81/2) t2
Now you can solve
for t!
-257.85 = –
(9.81/2) t2
t2 =
52.569
t ≈ 7.25 s
Thatfs the time it
takes terebi-san to smash into the ground. To get his
initial velocity, go back and plug t into the x equation we had after solving
for vo.
vo = 207.85/t
vo = 207.85/7.25
vo
≈ 28.7 m/s
Yay! Pointless answers to a pointless, relatively
retarded problem. But hey maybe you learned somethingc-_- or notcc..
Eh heh.
This is why I
donft teach physicsc
OH OH OH OH !!!!!! But firstcHerefs a
fun continuation of the problem!!!!
Assume that hide-sama takes 1.5 s to actually accelerate the TV from rest before
it leaves his hands to continue the journey down to crash land.
Given that the TV
is 20 kg, figure out what force hide threw the TV with.
Herefs the magic
equation,
Wefre finding force, we know the masscbut what about acceleration? O_o
Well, acceleration
is the change in velocity per unit time. (if youfve
had calculus: dv/dt)
The TV goes from 0
m/s to our previously found ginitialh velocity of 28.7 m/s so that makes the
change in velocity:
v - vo
= 28.7 – 0 = 28.7
Now it goes from 0
to 28.7 m/s in the space of 1.5 s, so that means that our change in time is
1.5.
So our
acceleration is:
a = dv/dt = 28.7/1.5 ≈
19.13 m/s2
NOW you can plug
this into F = ma, along with the given mass of 20kg to get:
F = ma
F = 20 ( 19.13)
F ≈ 380 N
THATfS ALL
FOLKSSS!!!!!!