TN2612 - introduction to special relativity
Lambert van Eijck
NPM2, dep. Radiation Science and Technology, fac. Appl. Sci, TU Delft
2018-2019
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
1
Course info
Teacher: Lambert van Eijck, l.vaneijck@tudelft.nl
Lectures: 1+8 colleges, 1+8 werkcolleges
Book: Dynamics and Relativity, Ch 11 t/m 14
Author: W.D. Comb
Publisher: Oxford
Examination: written
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
2
today: classical mechanics as context for special relativity
Figure: Galileo Galilei (1564-1642) in Pisa: which object falls faster: the
light or the heavy object? Defines speed, acceleration, gravitational
acceleration.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
3
a typical success story of classical mechanics: dynamics of
many particles
Figure: i particles with mass mi , position xi , yi , zi , velocity v~i .
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
4
dynamics of many particles
Figure: these i particles can be stars, planets, steel balls or atoms!
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
5
the most famous person in classical mechanics is Isaac
Newton (1642-1727)
~ = m · ~a
F
In the case of hard spheres, the trajectories of the individual
particles are described as: ~xi (t) = ~xi0 + ~vi0 t + 0.5~ai t 2 and the
~ = m · ~a
interactions are described using: F
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
6
center of mass
The center of mass of a system of particles:
~xi (t) = ~xi0 + ~vi0 t + 0.5~ai t 2
is defined as:
~ cm =
X
P
xi
i m·~
P
i m
Lambert van Eijck
The center of mass is a position defined
relative to an object or system of objects. It is
the average position of all the parts of the
system, weighted according to their masses.
For simple rigid objects with uniform density,
the center of mass is located at the centroid.
there should be an i subscript for m too!!
TN2612 - introduction to special relativity - lecture 0
7
velocity of the center of mass
~ cm =
Similarly, the velocity vectors ~vi can be added: V
Lambert van Eijck
P
vi
i mi ·~
P
i mi
TN2612 - introduction to special relativity - lecture 0
8
defining an inertial frame of reference:
~ cm (t) = X
~ cm (0) + V
~ cm · t
X
position of center of mass at time t is its initial position (t=0) + its
speed times time passed
the second derivative of displacement x is
acceleration a
P
m ·~ẍ
The acceleration of the center of mass aCM is: ~acm = Pi mi i i We
i
imply that there are no external forces and:
the sum of all m for i is equal to the total
P ~
mass of the system
i Ẍcm = 0 the total acceleration of the center of mass is 0
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
9
inertial frame of reference for 2 particles:
~ cm (t) = X
~ cm (0) + V
~ cm · t
X
Figure: Two particles (red) trajectory can be ”summarized” as one CM
system (green).
For the two particles system i, j: mi ~ẍi + mj ~ẍj = 0
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
10
relative velocities in two reference frames
that's because you move with it, so there
is no relative motion between you and it
If you decide to ’sit’ on the CM reference frame, then VCM = 0 for
you. The relative velocities ∆v21 of the (red) particles 1 and 2
0 ,
according to me should be the same as the relative velocity v21
according to you.
~v21 =
0
~v21
Lambert van Eijck
=
∆~x2
∆t
∆~x20
∆t 0
−
−
∆~x1
∆t
∆~x10
∆t 0
relative velocity is described as the difference in
velocity between one particle (Dx2/t) and the
other (Dx1/t)
for a different observer, these relative velocities
are going to be different, but the difference
between them should be the same
TN2612 - introduction to special relativity - lecture 0
11
relative velocities in two reference frames
~v21 =
0 =
~v21
∆~x2
∆t
∆~x20
∆t 0
−
−
∆~x1
∆t
∆~x10
∆t 0
so,
~v21 =
0 =
~v21
Lambert van Eijck
(x2 −x1 )tB −(x2 −x1 )tA
tB −tA
(x20 −x10 )t 0 −(x20 −x10 )t 0
B
tB0 −tA0
A
TN2612 - introduction to special relativity - lecture 0
12
relative velocities in two reference frames
~v21 =
0 =
~v21
(x2 −x1 )tB −(x2 −x1 )tA
tB −tA
(x20 −x10 )t 0 −(x20 −x10 )t 0
B
tB0 −tA0
A
Although positions x might be unequal for us (x 6= x 0 ), in general,
any distance (= relative positions!) ∆x = ∆x 0 is the same for you
and for me. This, however, implies that we have determined these
distances at tA = tA0 and tB = tB0 and so ∆t = ∆t 0 .
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
13
~ = m · ~a
F
Figure: Classical mechanics is based on standards of length and time
(which we will have to sacrifies for Special Relativity).
~ = m · ~a are so successful that we (you)
Classical mechanics and F
use it still today, on a daily basis. It is based on the notion of
distance and time intervals that seem ’set in stone’.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
14
inertial frame of reference for 2 particles:
~ cm (t) = X
~ cm (0) + V
~ cm · t
X
Figure: The collision of two particles can be described in the CM system.
How to describe the collision of the two particles system 1, 2 with
mass m (Christiaan Huygens)?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
15
~ = m · ~a
F
I
I
I
I
I
describe the system in an inertial frame of reference
~ is needed to change the motion of mass m
force F
~ and ~a are vectors, implying addition
F
P ~
P ~
a or equivalently:
so rather:
i Fi = m · ~
i Fi =
d~p
dt
momentum:
p = m*v
F = dp/dt
intuitively, force causes acceleration, but does the formula
imply such causality?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
16
Exercise 1
Figure: Wikipedia: ”The froghopper can accelerate at 4000 m/s2 over 2
mm as it jumps”
I
I
I
what force is directly responsible for the acceleration of the
froghopper?
How big is this force if the acceleration is vertical and the
insect weighs 0.1g?
What is the height that the froghopper reaches?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
17
Exercise 2
Usain Bolt gives you a ’discount’ of 20m on a 100m running
competition (you start at x = 20m in front of him).
I
What average speed do you need to prevent him from winning
(assuming he is as fit as ever)?
You lost anyway, so Usain offers you to cross the starting position
x = 0 at your full speed.
I
What average speed do you need to run, to win?
I
What is the distance you’ve run, according to you?
I
What time did it take you to win this sprint, according to your
watch?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
18
force and energy ”before” the era of E = mc 2
R
~ · dx
~
work done by a force is defined as: W = F
2
kinetic energy is defined as: ∆K = m · v /2
potential energy is defined as: U = k · (∆x)2 /2 or U = mgh, or ...
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
19
Exercise 3
An intergalactic rocket with kinetic energy K explodes into two
pieces, each of which moves with twice the speed of the original
object.
I
determine the change in direction of each of the pieces w.r.t.
the flight direction prior to the explosion.
I
calculate the ratio of the internal and center-of-mass energies
int
.
after the explosion KKCM
I
which conservation laws did you use for your calculations?
I
which conservation laws did you not use for your calculations?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
20