The shutter speed might be 1/200, but the flash duration will probably be quite a lot shorter than that (1/1000 or thereabouts maximum), especially when used at low output.
Therefore the best thing to do is to make sure that the ambient light is insufficient to create a blurred image, and just rely on the flash exposure.
Most portable flashguns control their output by shortening the flash duration, so the closer the flash is to the object the shorter the duration of the flash - down to around 1/10,000 to 1/30,000 second, depending on which flash you get. Most manufacturers give that in the spec.
Non-automatic, studio flashes may not use time as the output control - in fact they may have a longer flash duration at lower output.
In general, continuous light is totally inappropriate for high speed still photography. Just suppose that you have a flashgun with a 200 Ws output (eg the Metz 60 series). At 1/64 power it has a flash duration of 1/5500 sec and an output of about 3 Ws. As that lasts 1/5500 s, it is like a 17 kW light during that brief period. Even if you used a continuous light source that was twice as efficient as the flash tube, that would be around 9 kW. 9 kW of high efficiency continuous light is very expensive. Way more than a second-hand Metz 60CT series for $150 to $200. If you used incandescent lamps, you would need about 30 kW. These are just ball-park estimates to give you an idea of the magnitude.
The Metz 60, though powerful, might not be the best for high speed use, by the way. I just used it because I have the numbers at hand. It has a 1/200 s flash at full output, so you need to reduce the output quite a lot to get a short burst like 1/6000 s. It would probably be better to start with a flash that has a short burst at full output, so you only need about 1/8 output to get to 1/6000.
Best,
Helen