Why do I feel funny in my tummy when riding a virtual roller coaster?
Why can I feel this, when no forces are actually being applied to my
  stomach?


Acceleration sensors are in the ears and are part of the vestibular system. The vestibular nuclei in the brain uses acceleration- and visual signals to decide what happens to your body. It can be fooled by visual signals without the presence of acceleration signals. The role of the vestibulo-autonomic reflex - you feel in your stomach - is probably to protect your inner organs from the damage they can possibly suffer by acceleration.

The same can happen by watching films with fast camera movements on a monoscopic display (e.g. normal films in the cinema or video games on an average LCD monitor, etc...), however not so often than by stereoscopic displays (e.g. 3d films in the cinema, stereoscopic video games on a head-mounted display like Oculus Rift, etc...). This is because stereoscopic displays create the perfect illusion of depth (e.g. they can even be used by a therapy to overcome acrophobia), while monoscopic displays don't. So your vestibular nuclei is fooled much easier by stereoscopic displays.

References:


  These data show that a subset of PBN/KF neurons, whose activity is
  altered by a nauseogenic stimulus also respond to body motion and that
  irritation of the stomach lining can either cause an amplification or
  reduction in the sensitivity of the units to vestibular inputs. The
  findings imply that nausea and affective responses to vestibular
  stimuli may be modified by the presence of emetic signals from the GI
  system.



2011 - Integrative responses of neurons in parabrachial nuclei to a nauseogenic gastrointestinal stimulus and vestibular stimulation in vertical planes
1995 - Study on vomiting related to the vestibulo-autonomic reflex.



  This study describes the relation between the vection produced by
  optical flow and that created by galvanic vestibular stimulation.
  Vection is the illusion of self motion and is most often experienced
  when an observer views a large screen display containing a translating
  pattern. This illusion has only limited fidelity and duration unless
  it is reinforced by confirming vestibular information. Galvanic
  vestibular stimulation (GVS) can directly produce the sensation of
  vection.



2005 - Virtual acceleration with galvanic vestibular stimulation in a virtual reality environment



  Neurons in the vestibular nuclei are not only activated by vestibular
  signals, but also by visual information. Consequently, the sensations
  of body motion derived from visual and vestibular signals are
  indistinguishable.
  
  Sensations of body movement can also be generated by visual signals
  arriving at the vestibular nuclei. During unnatural movements of the
  body, conflict between visual and vestibular information leads to
  disorientation and nausea, as in the oculogyral illusion, the Coriolis
  effect, and vection.



Foundations of Perception - The body senses - The vestibular system



  Three-dimensional movies presented via stereoscopic displays have
  become more popular in recent years aiming at a more engaging viewing
  experience. However, neurocognitive processes associated with the
  perception of stereoscopic depth in complex and dynamic visual stimuli
  remain understudied. Here, we investigate the influence of
  stereoscopic depth on both neurophysiology and subjective experience.
  Using multivariate statistical learning methods, we compare the brain
  activity of subjects when freely watching the same movies in 2D and in
  3D. Subjective reports indicate that 3D movies are more strongly
  experienced than 2D movies. On the neural level, we observe
  significantly higher intersubject correlations of cortical networks
  when subjects are watching 3D movies relative to the same movies in
  2D. We demonstrate that increases in intersubject correlations of
  brain networks can serve as neurophysiological marker for stereoscopic
  depth and for the strength of the viewing experience.



2014 - Stereoscopic depth increases intersubject correlations of brain networks



  Statistical differences in theta activity showed that the real and 3D
  environments caused similar cognitive processes, while the 2D caused
  an increase of anxiety indicating that perhaps participants were
  looking for the third dimension. Beta and gamma activity showed that
  participants perceived the third dimension of the stereoscopic
  environment as in the real one, something that did not happen in the
  2D environment. Our findings indicate that stereoscopic 3D virtual
  environments seem to approximate the real ones as far as it regards
  the cognitive processes they cause. Three dimensional stereoscopic
  environments increase users’ attention over the 2D and cause less
  mental effort.



2013 - Stereoscopic perception of women in real and virtual environments: A study towards educational neuroscience
2001 - Human Cortical Activity Correlates With Stereoscopic Depth Perception



  In this study the effects of virtual reality exposure therapy (VRET)
  were investigated in patients with acrophobia. Feelings of presence in
  VRET were systematically varied by using either a head-mounted display
  (HMD) (low presence) or a computer automatic virtual environment
  (CAVE) (high presence). VRET in general was found to be more effective
  than no treatment. No differences were found in effectiveness between
  VRET using an HMD or CAVE. Results were maintained at 6 months
  follow-up. Results of VRET were comparable with those of exposure in
  vivo (Cyberpsychology and Behavior 4 (2001) 335). In treatment
  completers no relationship was found between presence and anxiety.
  Early drop-outs experienced less acrophobic complaints and
  psychopathology in general at pre-test. They also experienced less
  presence and anxiety in the virtual environment used in session one as
  compared to patients that completed VRET.



2004 - Treatment of acrophobia in virtual reality: The role of immersion and presence



  Despite the increasing use of virtual reality, the impact on cerebral
  representation of topographical knowledge of learning by virtual
  reality rather than by actual locomotion has never been investigated.
  To tackle this challenging issue, we conducted an experiment wherein
  participants learned an immersive virtual environment using a
  joystick. The following day, participants' brain activity was
  monitored by functional magnetic resonance imaging while they mentally
  estimated distances in this environment. Results were compared with
  that of participants performing the same task but having learned the
  real version of the environment by actual walking. We detected a large
  set of areas shared by both groups including the parieto-frontal areas
  and the parahippocampal gyrus. More importantly, although participants
  of both groups performed the same mental task and exhibited similar
  behavioral performances, they differed at the brain activity level.
  Unlike real learners, virtual learners activated a left-lateralized
  network associated with tool manipulation and action semantics. This
  demonstrated that a neural fingerprint distinguishing virtual from
  real learning persists when subjects use a mental representation of
  the learnt environment with equivalent performances.



2009 - Impact of the virtual reality on the neural representation of an environment



  Our main results are (1) stereoscopy in games increased experienced
  immersion, spatial presence, and simulator sickness; (2) the eff ects
  strongly differed across the three games and for both ge nders,
  indicating more affect on male users and with games involving depth
  animations; (3) results related to attention and cognitive involvement
  indicate more direct and less thoughtful interactions with
  stereoscopic games, pointing towards a more natural experience through
  stereoscopy.



2012 - Understanding User Experience in Stereoscopic 3D Games



  The results indicate that in a 2D movie viewers tended to look at the
  actors, as most of the eye movements were clustered there. The
  significance of the actors started at the beginning of a shot, as the
  eyes of the viewer focused almost immediately to them. In S3D movie
  the eye movement patterns were more widely distributed to other
  targets. For example, complex stereoscopic structures and objects
  nearer than the actor captured the interest and eye movements of the
  participants. Also, the tendency to first look at the actors was
  diminished in the S3D shots.



2014 - What do people look at when they watch stereoscopic movies?



  For the correlation between anxiety and presence, the results show
  very low correlation between anxiety and presence.



2009 - Using augmented and virtual reality for the development of acrophobic scenarios. Comparison of the levels of presence and anxiety



  Stereoscopic visualization in cinematography and VR creates an
  illusion of depth by means of two bidimensional images corresponding
  to different views of a scene. This perceptual “trick” is used to
  enhance the emotional response and the sense of presence and
  immersivity of the observers. An interesting question is if and how is
  possible to measure and analyze the level of emotional involvment of
  the observers during a stereoscopic visualization of a movie or of a
  virtual environment. The final goal of this research is a challenge,
  due to the large number of sensorial, physiological and cognitive
  stimuli involved. In this paper we begin this research by analyzing
  eventual differences in the brain activity of subjects during the
  observation of monoscopic or stereoscopic contents. To this aim, we
  have performed some experiments collecting EEG data using a
  Brain-Computer Interface device from two groups of users, during the
  observation of stereoscopic and monoscopic short movies inside the
  Virtual Theater of the University of Milan. From the analysis of the
  collected data, it seems that interesting differences are present in
  the average brain activity among the observers in the two groups, with
  a significative effect of stereoscopic visualization.



2012 - Analysis of brain activity and response during monoscopic and stereoscopic visualization