In physics problems, the earth is usually considered to be an inertial frame. The earth has a gravit

A ball is thrown at a relative velocity of $3m/s$ in $+ve$ $x-$ direction in a rocket in gravity free space. The rocket has a constant acceleration of $2m/s^2$ also in $+ve$ $x-$ direction.
Whether uniform acceleration equations of motion would apply in above case when using the rocket as a frame of reference. As an example, would the equation $s=ut+.5a{t}^2$ apply if rocket is taken as frame of reference?

"Strictly speaking, Newton’s laws of motion are valid only in a coordinate system at rest with respect to the 'fixed' stars. Such a system is known as a Newtonian, or inertial reference frame. The laws are also valid in any set of rigid axes moving with constant velocity and without rotation relative to the inertial frame; this concept is known as the principle of Newtonian or Galilean relativity."
Why should the inertial frame of reference not spin?

For a car that is accelerating linearly, the non-inertial frame of reference is the driver in the car where from his reference frame, the car is stationary. It is so called stationary because the non-inertial frame of reference has the same acceleration as the car. Is like the car's acceleration "transform" the driver frame of reference into a non-inertial. That's why in the non-inertial frame of reference, there is no force acting on the car.
But when the car is driving in circles at a constant speed, in the non-inertial frame of reference there is a force acting on the car, which is the centripetal force. Why isn't this frame of reference like the above, not having the acceleration found in their each respective inertial reference frame? Why can't we have a non-inertial reference frame(due to rotation) whereby there is no centripetal force, subsequently eradicating the need for a centrifugal force?

The Fourier rate equation of heat conduction states:
$\overrightarrow{q}=-k\mathrm{\nabla }T$
But I'm wondering if this is valid in every frame of reference, because the heat flux $\overrightarrow{q}$ does obviously change when the area under consideration is moving. The equation is often applied to moving fluids, and because there is no objective way to determine a non-moving frame of reference in a moving substance, I guess that the equation is valid for every inertial frame of reference, but I just wanted to be sure.

A rotating frame of reference (since you rotate your BEC to generate these), where the energy is given by:
$\tilde{E}[\mathrm{\Psi }]=E[\mathrm{\Psi }]-L[\mathrm{\Psi }]\cdot \mathrm{\Omega },$
Where $\mathrm{\Omega }$ is the rotational velocity which you apply to the BEC.
Now the argument that the term $L[\mathrm{\Psi }]\cdot \mathrm{\Omega }$ should be substracted comes from the fact that in a rotating frame your system loses that fraction of rotational energy. Now I was wondering if anyone knew where the form of $L[\mathrm{\Psi }]\cdot \mathrm{\Omega }$ came from?
I know that in a rotating frame of reference you have that $\overrightarrow{v}={\overrightarrow{v}}_r+\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{r}$ . If you fill this in into the kinetic energy and use some basic definitions, you get that the extra effect of rotation is given by:
$\frac{1}{2}I{\mathrm{\Omega }}^2=\frac{1}{2}J\mathrm{\Omega }.$
This yields half of the value that is used in the book, is there something that I'm missing or not seeing right ?

"You are traveling in a car going at a constant speed of 100 km/hr down a long, straight highway. You pass another car going in the same direction which is traveling at a constant speed of 80 km/hr. As measured from your car’s reference frame this other car is traveling at -20 km/hr. What is the acceleration of your car as measured from the other car’s reference frame? What is the acceleration of the other car as measured from your car’s reference frame?"

Shouldn't they both appear to have an acceleration of zero, because both velocities are constant? I can imagine sitting in the faster car and watching the slower car, its speed would not appear to change, only its position?

Can you identify the direction of your movement? Let's say ship moving nose forward, or tail forward? Is there a sense of "direction of movement" in such an inertial frame of reference?