Physics 105A
Projectile Problem.
2nd Order Linear Equations,
Simple Harmonic Oscillator
Damped Harmonic Oscillator
28 June 2016
Manuel Calderón de la Barca Sánchez
Newton’s Apple



Projectile is aimed straight at the apple.
Apple falls the moment projectile is launched.
Will it hit the apple?
ry
v0
q
rx
28 June 2016
MCBS
Newton’s Apple

If there is no gravity…
ì x =v t
ï r
0x
Projectile's position í
ïî yr = v0 y t
ì x =r
ï a x
Apple's position í
ïî ya = ry
Aiming at the apple:
ry
v0
q
rx
28 June 2016
MCBS
ry
rx
=
v0 y
v0 x
Newton’s Apple

If there is gravity
ì
xb = v0 x t
ï
Projectile's position í
1 2
ï yb = v0 y t - gt
2
î
ì
xm = rx
ï
Apple's position í
1 2
ï ym = ry - gt
2
î
Aiming at the apple:
ry
v0
q
rx
28 June 2016
MCBS
ry
rx
=
v0 y
v0 x
Newton’s Apple
 If the initial speed is large…
 projectile will hit apple at earlier time.
ry
v0
q
rx
28 June 2016
MCBS
Things to note:
 The term -½gt2 is absent if:
There is no gravity.
There is gravity, but we put the experiment in a freely-falling
elevator: change our frame of reference.
– Gravity produces the EXACT SAME acceleration on all objects!
 Important in General Relativity.
 An experiment done in a gravitational field g, or inside an elevator accelerating
with acceleration g will yield the same results.
 The person in the elevator cannot tell the difference.
28 June 2016
MCBS
Chapter 4: Oscillations

Linear Differential Equations
x = ax
x = ax
x + 2g x + ax = 0
28 June 2016
MCBS
So
who
is
coming
to
my
party?
Might stay at rest, but if there is a critical mass then
I’ll be forced to attend. –Isaac Newton
Been looking for some action, have a lot of energy.
–William Rowan Hamilton
Party? I’m buoyant! Will bring pi. –Archimedes
Right on! We’re radiating with enthusiasm! –The
Curies.
How special! It should be relatively easy to go, in
general. -Albert Einstein.
I’m electrified about it! -Alessandro Volta
Ok, but might not be up to current fashion. –André
Marie Ampére
I resisted at first, but I think I’ll go. –Georg Ohm.
Can’t, I’m under so much pressure I feel squashed
–Bob Boyle
Should be illuminating. -Thomas Edison
Sure, want to let off some steam. –Jimbo Watt
The party has very high potential. Might bring a few
gadgets, you got wireless? –Nikola Tesla
I’ll think about it. –René Descartes
My brother and I will go, if we can get a flight. –
Wilbur Wright
Be there on the dot. Can’t say more, gotta dash. –
Sam Morse
I’m uncertain if I’ll be there... might be somewhere
else. –Werner Heisenberg
Will miss, but I promise to attend with greater
frequency. –Heinrich Hertz
Your parties are pretty warped! I’ll try to string a
hole in my schedule. –Stephen Hawking
Dunno. Will see what evolves. –Chuck Darwin
Have to take my cat to the vet. Wait.. maybe not.
–Erwin Schrödinger
Potluck? I’ll mix some veggies together and see
what comes out. –Greg Mendel
Just heard, already salivating. –Ivan Pavlov
Absolutely cool! –Lord Kelvin
Things are in flux, but could be induced to go. –
Michael Faraday
O-eh o-eh! feeling hot, hot, hot! –Jimmy Joule
Simple Harmonic Motion

Hooke’s Law: F=-kx.
x = -w x
2
28 June 2016
MCBS