Normally we would be concerned with starting from some fundamental axioms and deriving a set of equations which allow us to handle real-world problems. We will skip about 90% of this,and work on a "need-to-know" basis. If you want to see a more systematic derivation, look here.
We want to understand the following
Distance, Velocity and Acceleration
Force
Newton's Laws
F = ma
Gravitational Force
Potential energy
Rockets
We start by describing motion
e.g. Achilles and the tortoise ": he wants to catch a tortoise that is 100 m away.
He runs twice as fast. Can he catch it?
So does he catch it?
We have an infinite series of moves
1
1 + 1/2 = 1.5
1 + 1/2 + 1/4 = 1.75
1 + 1/2 + 1/4 + 1/8 = 1.875
1 + 1/2 + 1/4 + 1/8 + 1/16 .......= 2
So we asked the wrong question: not how many moves does he take, but how far does he have to go.
Infinite series can have a finite sum!
What we need to know
Can we escape the earth's gravity?
Could we shoot a space capsule from a gun?
Can we escape the solar system?
Can we escape anything?
How do we have a stable orbit?
How fast can a rocket go?
Can we beat the constraints somehow?
Speed and Velocity
Average Speed is just distance/time
\color{red}{
Speed = \frac{d}{{t_1 - t_0 }}}
e.g. if Achilles runs d=100m starting at 4.00.00 p.m. and ending at 4.00.20 p.m., then
An action is a force exerted by one object on another
The reaction is the force exerted by the second on the first
Note that it is particularly easy to forget reaction forces: in this case,
if you ignore the reaction force, the block would fall through the table.
Gravitation
Most important (for us) is gravitational force, which gives rise to weight
To a good
approximation, all objects falling near the Earth's surface have the same acceleration
a = -g where g = 9.8 m/s2
How are grav. accn. and velocity connected?
A ball is thrown up in the air. During the first part of its motion (before it reaches its maximum height)
the velocity is upward and the acceleration is downward.
Hence Newton's 2nd. law shows that everything has the same acceleration in
a gravitational field.
We have the ingredients to understand the solar system
Why does the moon take 27.3 days to orbit the earth? (distance R ≈ 385000 km)
How does the moon stay up?
By falling!
faster
and faster
Law of universal gravitation
: applies between any two bodies anywhere in the universe
Applied to earth-moon: need to know mass of earth (M) and G (not mass m of the
moon, since it will cancel out),
but we do know force at earth's surface
\color{red}{
F = \frac{{GMm}}{{R^2 }}}
where RE is the radius of the earth, and all we really need is the product GM
P = 27.4 days (!!!!!!!!!!!)
Note the logical flow
This was the first time that laws deduced on the Earth were seen to apply outside!
Note that this imples that grav. force gets weaker as we move away from the earth
G=6.67x10-11 N m2 kg-2 is a universal constant. The first direct test of this was by measuring the deflection of a plumb-bob near a mountain in Scotland (Schiehallion)
Energy
Newton's second law gives us a relation between velocity and force, via \color{red}{\vec F = m\vec a}
If the force is complicated , then solving for the velocity can be very difficult.
Fortunately there is a better way: a new idea called Energy.
Kinetic Energy
defined to be
\color{red}{
T = \frac{1}{2}mv^2 }
for a particle with mass m, vel v.
Potential Energy
If you drop something, kinetic energy increases. This energy is originally in the form of potential energy (P.E.).
Total energy = P.E.+ K.E.= constant
Note that the force would be very complicated in this case, so one couldn't actually solve via F = ma
Need a new unit for energy
1 Joule= 1 kg m2s-2
(Joule originated study of heat energy --> mechanical energy)
Conservation of Energy
If the forces are conservative, then total (mechanical) energy will be conserved: it can be transformed from one form to another.
(P.E. ⇔K.E.)
Energy is always conserved, but we need to introduce more kinds:
Heat⇔chemical ⇔ potential ⇔ nuclear ⇔kinetic
Rocket
Mass continuously shot out of back: solving this gives
\color{red}{v = v_0 + V\ln \left( {\frac{{M_0 }}{M}} \right)}
note: a rocket does not need anything to push against!
Escape Velocity
How hard would you need to throw something so that it never came back?
At the earth's surface
\color{red}{
v = \sqrt {\frac{{2GM}}{R}} }
e.g. for the earth: R = 6500 km, G = 6.67x10-11, M= 6x1024 kg
\color{red}{v_{escape} = 11100{\rm{ms}}^{{\rm{ - 1}}} = 11.1{\rm{km s}}^{{\rm{ - 1}}} }
Conclusions/Consequences
Can we escape the earth's gravity?
Easy in principle: just need to launch something with a speed of 11000 m/s
Could we shoot a space capsule from a gun?
Yes, but we are limited by the accn we can stand:
For long periods, can manage 1.5 g easily
Short periods 4g
e.g. gun with barrel of 1 km, reaching escape vel -> a = 6200 g (!!!)
Can we escape the solar system?
escape vel = 42 km/s
Pioneer has already done it
Can we escape anything?
See later
How do we have a stable orbit?
Easy: just above atmosphere, orbital vel = 7.8 km/s
How fast can a rocket go?
Depends on exhaust speed. Suppose we have rocket with a mass of 100 tonnes, final mass 1 tonne, exhaust vel = 3000 m/s, final vel will be ~14000 m/s
BUT the 1 tonne must include the mass of the container
