WEBVTT
00:00:00.080 --> 00:00:02.821
This is College Physics
Answers with Shaun Dychko.
00:00:03.141 --> 00:00:05.832
We're going to find the ratio
of speeds of an electron
00:00:05.832 --> 00:00:09.221
to a negative hydrogen ion
accelerated to the same voltage.
00:00:09.883 --> 00:00:12.858
So, the change in kinetic energy
of each charged particle
00:00:13.010 --> 00:00:16.043
will be the opposite to the
change in potential energy,
00:00:16.232 --> 00:00:19.825
and the word opposite is indicated
with this negative sign there.
00:00:20.370 --> 00:00:24.000
So, a decrease in potential energy results
in an increase in kinetic energy.
00:00:24.443 --> 00:00:27.825
And, potential energy change is the charge
00:00:27.825 --> 00:00:30.683
times the potential difference
through which it travels.
00:00:31.410 --> 00:00:32.858
And that's *Delta V*.
00:00:33.890 --> 00:00:37.345
So, the change in kinetic energy can
also be thought of as one-half mass
00:00:37.345 --> 00:00:41.221
times final velocity squared minus one-half
mass times initial velocity squared.
00:00:41.483 --> 00:00:44.087
But we're told that the
initial velocity is zero.
00:00:44.087 --> 00:00:45.170
It starts from rest.
00:00:45.170 --> 00:00:46.596
And so, this term is zero,
00:00:46.916 --> 00:00:51.200
meaning that change in kinetic energy
is then one-half *M* *V* final squared.
00:00:52.712 --> 00:00:56.283
So, for the electron, we have one-half
times the mass of the electron
00:00:56.487 --> 00:00:58.880
times the final velocity
of the electron squared
00:00:59.229 --> 00:01:04.269
equals negative charge of the
electron times change in voltage.
00:01:04.538 --> 00:01:07.963
Now, I did not put a subscript E
for charge of the electron here
00:01:08.174 --> 00:01:10.370
because the charge of the electron
00:01:10.370 --> 00:01:15.141
and the charge for the negative
hydrogen ion are the same.
00:01:15.338 --> 00:01:17.454
They're both one elementary charge.
00:01:17.454 --> 00:01:19.760
And so, there's no need to
have a subscript there.
00:01:20.727 --> 00:01:23.461
And, they also go through the
same potential difference,
00:01:23.461 --> 00:01:26.887
and so there's no need for a
subscript for this *Delta V* either.
00:01:27.905 --> 00:01:29.520
And so on this line in green,
00:01:29.520 --> 00:01:34.305
we have the change in kinetic
energy for the hydrogen ion,
00:01:34.763 --> 00:01:40.581
one-half mass of the hydrogen ion times
the hydrogen ion's final velocity squared
00:01:40.792 --> 00:01:42.552
equals negative *Q* *Delta V*.
00:01:43.338 --> 00:01:46.843
Now, we divide these two,
and the one-halves cancel.
00:01:47.374 --> 00:01:53.854
And, we have *Me Vfe squared* over
*Mh Vfh squared* equals one.
00:01:54.240 --> 00:01:57.847
And, the one comes from this, the
right hand sides being divided,
00:01:57.847 --> 00:02:01.236
but they have the same factors
there, so this makes one.
00:02:02.778 --> 00:02:09.076
And, we'll solve for *Vfe squared* over *Vfh
squared* by multiplying both sides by
00:02:09.076 --> 00:02:12.000
mass of the hydrogen ion divided
by mass of the electron.
00:02:14.661 --> 00:02:20.509
And, we end up with *Vfe squared*
over *Vfh squared* is *Mh* over *Me*.
00:02:21.250 --> 00:02:22.960
And then, we'll take the
square root of both sides,
00:02:23.323 --> 00:02:29.978
to finally solve for the ratio of velocity
of the electron versus the hydrogen ion.
00:02:31.112 --> 00:02:33.229
And so, that's going to be the square root of
00:02:33.418 --> 00:02:36.036
the mass of the hydrogen ion divided
by the mass of the electron.
00:02:36.429 --> 00:02:40.341
So, that's square root of 1.67 times
ten to the minus 27 kilograms,
00:02:40.341 --> 00:02:45.229
mass of the hydrogen ion, divided by 9.11
times ten to the minus 31 kilograms,
00:02:45.229 --> 00:02:48.356
mass of the electron, which is 42.8.
00:02:48.865 --> 00:02:50.952
So, the velocity of the electron
00:02:51.069 --> 00:02:55.461
will be greater than the final velocity of
the hydrogen ion by a factor of 42.8.