Suppose you are in a closed box which is in motion 
Suppose you are inside a closed box which is moving with a constant speed with no acceleration whatsoever. You cannot see outside the box. If you are given all the instruments you need, what experiment could you perform to ascertain that you are, in fact, in motion?

This was a question asked in an interview for admission to a pure sciences research university.
The only thing I was able to think of was maybe if we somehow converted the mass of some object into energy it would be slightly more than what we would expect from $E=mc^2$ since it also has kinetic energy.
Is this even remotely correct? Does it even have an answer or was it simply a trick question?
 A: First of all the question is a little ambiguous , but I assume that it meant "constant speed" relative to  an inertial frame(Newtonian definition) if so , then there is  no experiment you can do (done entirely in the box) to figure out whether your moving with constant speed (for example relative to earth) or other things are moving and your fixed or it's a combination of both .
Note:I assumed the non-relativistic case , but even considering SR my answer wouldn't change .
A: You can't.  This is a literal textbook question about Galileo's ship.  I cannot understand why this seems not to be taught in schools, the idea is nearly 400 years old, and the description of it is only a couple of hundred words.
A: This question is basically asking about the principle of relativity, which states:

Physical laws should be the same in every inertial frame of reference.

Since you're moving at constant speed, you are in an inertial frame of reference, and there is no detectable difference. That's why you are equally justified to say you are at rest and the rest of the universe is moving backwards at constant speed. This principle is also why we say there's no preferred frame of reference.
A: The interview question is inherently flawed because the box must permit its contents one path of interaction with the exterior: gravity. The reason is that other masses within the box's light cone, however remote, will accelerate the box and must accelerate its contents as well. The contents is supposed to move with the box along the same geodesic. The box can not, even not within the realm of our thought experiment, shield its contents from gravity in the way we can assume it shields from radiation.
Now the gravitational field will be ever-so-slightly inhomogeneous; there will be tidal forces within the box which can be measured. The universe is supposedly isotropic when perceived from the cosmic rest frame. If you start moving relative to it, you'll have anisotropic inhomogeneities in the gravitational fields you measure. For example, if the box is large enough to build a Ligo in it you'll measure a red/blue shift of the gravitational wave frequencies from large mass collisions. Even if, to make it harder, we suspend such events for the duration of our thought experiment you still have "inhomogeneity events", just ones which are harder to measure (passing a far star etc.). These events will be anisotropic while you move relative to the cosmic rest frame.1
The only way to prevent gravity from betraying your movement is to imagine the box in an empty light cone. Then there are two questions left to ask the interviewer:

*

*What exactly was the box supposed to shield me from again?

*Which motion?


1 Of course, the gravity from far objects will be weak, their inhomogeneities minuscule, and unless the box is moving at cosmological speed their anisotropy will be so small that only the keenest and most patient experimenter will be able to obtain statistically significant data through the eons. But hey, it's a thought experiment. We are not putting people in boxes.
A: As other's have said, there's no experiment you can do to determine your velocity.

maybe if we somehow converted the mass of some object into energy it would be slightly more than what we would expect from $E=mc^2$ since it also has kinetic energy.

No, that won't work. The object is (presumably) at rest in the frame of the box, so it has zero kinetic energy relative to that frame. There's no such thing as absolute kinetic energy or velocity, it's always measured relative to some frame.
But if you have, for example, an electron and a positron, which both have (rest) mass of $511\, keV/c^2$, and those particles have a total kinetic energy of $E$ relative to the box, when you annihilate the electron  with the positron the gamma rays produced will have total energy of $E + 1022\,keV$.
A: There always exists an inertial reference frame in which your velocity is not identically zero, so in some sense, you could say you are "in motion" in any situation. However, since the box is moving inertially, it would often be conventional to set its velocity to zero.
A: More heat would be generated on different sides of the box from friction. So assuming they tell you the only thing that can be going on outside is movement or not, that would give an answer. Unless you are in a vacuum.
A: It can be done depending on the circumstances
We need to know (1) where you are relative to the nearest massive object (2) what the constant 'speed' is and its direction (3) how long you have to carry out the experiment (4) if the box is allowed to rotate.
For example
Suppose the box starts on the surface of the Earth, and moves in a straight line, tangent to the surface. You will leave Earth. As you do so, the apparent direction and force of gravity will change. A simple static pendulum will detect one and a spring balance will detect the other.
A: There are a few ways you could attempt to answer the trick question and thereby give the interviewer an idea of how you approach problems (after you've first said "under our current understanding of physics, there is no privileged inertial reference frame" and thereby answered the object-level question).
One of the instruments you need might be a drill or an X-ray-sensitive plate, for example, so that in a couple of different ways you can start to see outside the box. Maybe one of the instruments is an odometer of some sort (and discuss how you might be able to make an odometer for e.g. a car that is on a freely-moving conveyor belt).
