Shutter speed refers to how long the shutter of the camera is open for. The longer it is open for the more blurry an object in motion will appear. A fast shutter speed will allow you to "freeze" a fast paced object, for example a running dog, allowing you to capture it in focus.
Let's look at some examples of what the same set up would look like shot at different shutter speed values. For this I have got Millie to run in front of me. I have set my camera to Shutter priority mode as this is the setting that I am wanting to control. The lens I used was my 70 - 200 zoom lens.
The following image was shot with a shutter speed of 1/6400, as you can see from the close up the image is pin sharp. The high shutter speed has "frozen" the action of Millie running.
Shutter Speed: 1/6400
This image has a shutter speed of 1/1250. As you can see, the main image looks in focus from a distance, but when you zoom into the image the motion blur is a lot more visible.
Shutter Speed: 1/1250
And finally this image was shot using a shutter speed of 1/500. Millie is completely out of focus from the motion blur.
Shutter Speed: 1/500
The camera's shutter work to open to let light fall onto the image sensor. The speed of these shutters determines how long they remain open. Shutter speed measures the length of time in seconds for how long the shutters are open. This is what give cameras their distinctive sound.
Shutter priority mode is useful when you are trying to capture a moving object. It allows you to have more control of enabling to freeze the motion or to blur it.
On most cameras you have the option to change the priority mode, even no point and shoot cameras! This mode will usually be marked as "S" or "Tv" on the menu.
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