Strictly, dynamic pressure is half the air density times the speed squared.

I don't know much about aeronautics, so let me work out the physical meaning of these units. This looks a lot like the classic kinetic energy equation, 1/2 * m * v^{2}, with air density substituted for mass to give units of energy per volume. So dynamic pressure is just the kinetic energy density of the air in joules (J) per cubic meter.

The basic units of J/m^{3} are the same as the pascal (Pa), the SI unit of pressure. And multiplying that pressure by an area (like the effective cross section of an airplane fuselage) gives the dynamic force, which in SI units would be newtons (N).

Did I get all that right?

I've always seen the equation for dynamic pressure as resembling the momentum equation, m*v, rather than the kinetic energy equation. If we take the air density times the velocity, we have the mass flow rate per square meter. That is,

rho * v

^{2} = (rho * v) * v = m-dot * v

So the equation gives us momentum per second per square meter. Since momentum per second is force, we have force per square meter, or pressure.