prove the Liouville's theorem?

a research document concerning a physical principle or a common technology is required.

I choose to do roller coasters.

Please describe physical principles on how roller coasters work, in depth please.

Links to graphs would be amazing

Enough information to make a 5 page researched paper, thank you

A crystal composed of atoms of one unknown species has experimentally determined values of molar heat capacity at
constant pressure and two different temperatures:

Cp (200 K) = 21.573 J.K^-1.mol^-1
Cp (100 K) = 13.360 J K^-1.mol^-1

Determine the approximate molar heat capacity at a constant volume of this crystal as a function of temperature over the interval from
50-300 K and plot it with the two experimental points.

State five physical characteristics of planet Earth’s structure.

A 65 g ice cube at 0 °C is placed in an insulated box that contains 8 g of steam at 200 °C. What is the equilibrium temperature reached by this closed system? Approximate the specific heat capacity of steam to be 1.5 J/g oC. All the other values can be found on the textbook. [Note: Assume that all of the steam condenses.]

What are photonic Crystal's, what are photonic bandgaps and what was Yablonovitch's work on the photonic bandgap.

A block of mass m₁ = 3.54 kg on a frictionless plane inclined at angle θ = 26.5° is connected by a cord over a massless, frictionless pulley to a second block of mass m₂ = 2.41 kg hanging vertically (see the figure). (a) What is the acceleration of the hanging block (choose the positive direction down)? (b) What is the tension in the cord?

When an airplane communicates with a satellite using a frequency of 1.535 X 10^9 Hz, the signal received by the satellite is shifted highr by 207 Hz. (a) Is the airplane moving toward or away the satellite? Explain. (b) What is the component of the airplane's velocity toward or away from the satellite?

In light of quantum mechanics and relativity theory, pragmatism is the best approach to understanding the nature of reality than either materialism or idealism. is this correct?

Four charges of magnitude +q are placed at the corners of a square whose sides have a length d. What is the magnitude of the total force exerted by the four charges on a charge Q located a distance b along a line perpendicular to the plane of the square and equidistant from the four charges? Ans: bqQ/πε 0 (b2 + d2 / 2) how to get this answer?

Now assume that Alice and Bob are twins, and Alice left Earth and Bob stayed behind fixing his spaceship. If Alice spent some time moving near the speed of light before returning to Earth, which statement is correct when Alice returns to Earth?

a. Alice will say that she is younger than Bob, and he will agree.

b. Bob will say that he is younger than Alice, and Alice will say that she is younger than Bob.

c. Alice will say that she is younger than Bob, and he will disagree.

For both resistances, plot the discharge curve of ln(V/V0) vs. time. From the slope of the line, calculate C (capacity). Please explain in detail how to plot/determine some of the values of ln(V/V0), how to determine the slope of the graph, and if possible demonstrate using Excel. Thank you.

Select a commercial antifreeze and for your own car theoretically calculate the amount (volume in oz) of antifreeze needed to prevent freezing of water in car’s cooling system at -34.0 ^OF.  Then compare this value with the one recommended by manufacturer of antifreeze. To be able to make simple calculation of the amount of antifreeze it is assumed that ethylene glycol, CH₂OHCH₂OH, is the only component of commercial antifreeze. (I have a 2016 Ford Raptor)

You need to know and do.

1. Use Freezing point depression formula
2. Need to know how to convert temperature
3. Need to know metric to British and vice versa conversion
4. Need to know the density of the antifreeze
5. Need to know the volume capacity of the car’s radiator

1. Derive each mathematical model of projectile motion and compare each part of derive, ( projectile motion only, projectile motion with wind force, projectile motion with wind force and projectile motion with wind force and air resistance) Note of sign of velocity Y as indicate go up or down.

(a)projectile motion
Derive the mathematical model (in 2D) using Newton’s 2nd Law and produce the ODEs (ordinary differential equation) involving the horizontal and vertical velocities respectively. Solve for the velocities using analytical methods and integrate the velocity functions to obtain the position functions. Parameters to consider include projectile mass, gravity, initial launch velocity and angle.

(b)projectile motion with wind force
Re-derive the math model adding in a wind force function, w(t), acting on the projectile during its trajectory. Considering an arbitrary w(t) that can be defined by the user.

(c)projectile motion with wind force and air resistance
Re-derive the math model by considering air-resistance and solve for the trajectory using the methods described in (b).

please answer complete answer 3 part and show the different