Can't shoot at 1/4000 sec?

I have a Nikon d5300 with 18-55mm and 70-300mm lens. I wanted to capture water droplets, I tried using a flash but I get a black background on the pic. My camera doesn't have the option to change the flash sync speed. I can only shoot at 1/250 using flash. What do I do so that I can click on 1/4000sec.

You can't shoot flash at high shutter speeds on an SLR because the shutter doesn't expose the entire film surface at the same time.

Up your ISO and live with the noise.

The maximum flash sync speed of the D5300 is 1/200s even with i-TTL compatible flash guns.

Do it in the dark. Leave the lens completely open (B setting, you need a cable release) Hold the flash in your hand and set it off. The duration of the flash is around 1/100000 of a second. Effectively, this gives you a shutter speed of 1/100000 of a second. The darker, the better.

Don't use the flash. Use a really bright steady light source.

1/4000 second shutter speed is borderline for water droplets. What you want to do is use a flash (off camera is better so you can adjust lighting angle) at its lowest power setting. At the lowest power setting, the flash duration will be around 1/20000 seconds, enough to stop water droplets. You need to use a thyristor flash which modern speedlights are. Set up your camera to normal sync speed, usually under 1/200 seconds. If you want to include background things, use a slower shutter speed. You need to operate your camera in manual mode to do this type of work.

You can't shoot flash at high speeds on nearly all SLR cameras, for good technical reasons connected with the focal plane shutters. Just be thankful that the top flash speed is around 1/160th to 1/250th, depending on model. In the good old bad old days it was 1/30th or 1/60th.

If the background is dark you need actually to use a lower speed to bring the "daylight" up to the power of your flash, so try 1/60th or 1/30th or slower. If you do not believe that, try shooting without flash at 1/4000th or even 1/1000th with all other factors (ISO, Aperture) held just as for the flash - you will certainly get a very dark picture with no useable image at all.

Most DSLRs can't sync past 1/250th of a second (unless the your camera has a leaf shutter in the lens barrel like my old Kowa SetR II SLR camera could sync at 1/500th sec when SLRs could only syn at 1/60th sec)

For best results, use sunlight and reflectors or a constant light source (hope the light doesn't flicker or you may see that effect in your image).