# The shutter speed rule

## The simple description

Use the inverse of the focal length as the minimum shutter speed to avoid blurred pictures (i.e. 50mm -> max 1/50 sec)

## "Executive summary"

There is a well known rule of thumb in the photographers' world that you shouldn't use a shutter speed lower than the focal length. This rule is coming from the 35mm film world. With such a camera indeed, it is recommend to shoot at for instance at least 1/200sec or faster when shooting at 200mm, and that to avoid any motion blur due to your own (lack of) stability. This is of course valid for hand held shooting only.

The usual translation in the digital world for APS-C sensors is that you should take the crop factor into account (1.5). This means that a 200mm lens will give you a field of view equivalent to a 300mm on a 35mm camera and hence will need a shutter speed of 1/300 sec or faster.

This is more or less known to several generations of photographers, but this is more than a rule of thumb... you can show why this is true with some basic mathematics and optics... But there is more to it. Sensor resolution is also playing a role and if you are a pixel peeper, looking at your pictures at pixel level and printing huge posters, you'll see than the better the camera (the higher the resolution), the higher the needed shutter speed.

## The boring explanation

First of all, my deepest apologies to those not familiar with mathematics or physics…

But the conclusions are very interesting…. (see at the end)

I did some serious thinking to understand the rule of thumb of using a shutter speed equal to the inverse of the focal distance.

Here are the results: first some (half-serious) maths:

To simplify the calculations, we assume we can model the lens with only one convex element with a focal distance f. We also assume that we focus on infinity and the picture is forming on the sensor at a distance corresponding to the focal distance.

Let's call

h = the height of the sensor (16mm for a Nikon D70 for instance, a 1.5 crop sensor camera)

v = the vertical resolution of the sensor (2000 for a Nikon D70 for instance, a 6MP camera)

s = shutter open time in seconds

The angle of view can easily be computed like this:

$LaTeX: a = 2 \arctan \frac{h}{2f}$

Examples:

50 mm lens on a D70: 2 arctan (16/100) 18°: vertical angle of view

Usual definition (taking the diagonal size into account)

50mm lens on a D70: 2 arctan (28.8/100) = 32° diagonal angle

50 mm lens on an F100: 2 arctan (43.2/100) = 46.7° diagonal angle (full frame or film format)

If we assume that the blur due to your own movements is depending on the angular velocity of your movement (you usually use the left hand to hold the lens at a certain distance from the sensor, and the longer the lens, the larger the movement).

The maximum angle variation during the shot should correspond to maximum one pixel on the sensor:

d(a) = arctan (h/(vf)) approx= h/(vf) for small angles

If you induce a movement resulting in an angular velocity of w, you need a shutter time s so that w x s <= d(a) (in other words, you have to shoot fast enough so that the angular movement you generate is less than the angle variation corresponding to a pixel on the sensor).

The conclusion :

Combining the formulas I get s (shutter time in seconds) = d(a)/w = (h/v) * w * (1/f) or in other words s = constant * (1/f) which is the well known thumb rule: HOWEVER, that constant factor depends on three factors:

- the lower your own angular velocity (shake) the lower the allowed shutter speed (evident)

- the shutter time should be proportional to the height of the sensor (h) – this is also known: DX sized (16mm) vs FF-35mm (24mm) means the safe shutter speed should be multiplied by the crop factor, 1.5

- the shutter time should also be adapted based on the RESOLUTION of the sensor (v) If linear resolution is increased by a factor 2, the safe speed should be as well: to be clear if 1/100sec is safe on a D1, you NEED 1/200sec on a D2X to get a sharp picture at pixel level.

Disclaimer: there are some approximations in the reasoning, but the qualitative conclusions are valid.

I will try to rephrase the conclusions using normal words:

The concept of a "safe shutter speed" based on focal distance is scientifically correct: (to compensate motion blur due to the photographer)

100mm means 1/100sec on a FF or 35mm camera

400mm means 1/400sec etc...

but the following elements also have to be taken into account:

- your own "shaking" ability - evident

- the crop factor

100mm means 1/100sec safe on an F100 BUT

100mm means 1/150sec safe on a D70 or D200 or D2X...

- the sensor resolution

1/100sec just safe (for you) on a D1 means 1/200sec (also for you) will be needed on a D200 or if 1/100 is safe on a D70 you'll need 1/125 on a D200 otherwise you'll get the impression that your pictures are not too sharp.

Hope this makes the point a bit more understandable.