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Say I have a spaceship, and I'm flying it through space at a constant high velocity V. The spaceship is pretty large (big surface area). In space however, there is cosmic dust scattered throughout (density of 10-6 × dust grain/m3 and mass 10-17 kg according to http://en.wikipedia.org/wiki/Cosmic_dust). So therefore with high enough surface area and velocity, I would be hitting quite a few at really high speeds. Would this hitting impart heat energy on my spaceship? (my intuition says yes, but I'm really not sure why, I'd guess because of friction, or change in kinetic energy?) If so, how would you calculate how hot you would get?

QCD_IS_GOOD
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As a first approximation work out how much mass is being intercepted by your ship per second. Multiply that by 0.5xV^2 and you will get the power in Watts (if V=metres/s) being dumped into your ship assuming all the kinetic energy is dissipated as heat.