# The perfect Pancake Toss

## 1. Up and Down

Where h is the height we toss the pancake; v is the launch velocity; m is the mass of the pancake; g is the gravitational field of the earth 9.81 metres per second per second;

Which we can rearrange to

Putting some numbers in a respectable toss of 0.5m needs a launch velocity of 3 metres per second.:

##### A toss 0.5 m high needs a launch velocity of just over 3 metres per second.

If you want mph that is 7mph, but believe me it won’t stay up for an hour and it won’t go up a mile high, so it’s really a metres per second problem.

Once you get to 5 metres per second it’s going up over 1.5m and is likely to hit the ceiling.

At 1.5 metres per second it won’t really get out of the pan.

Now we can work out time of flight; it is simply:

So for our 0.5m throw it will be in the air for 0.6 seconds (just over half a second)

## 2. The Spin

We need the pancake to turn through half a turn, 180 degrees, or in mathematical terms that is π radians. It has to do that while it is in the air: this gives an equation for the angular momentum:

##### So for our 0.5 m throw we need to simultaneously impart an angular velocity of 4.9 radians per second.

You have to toss the pancake with an upward velocity of 3 metres per second and at the same time spin it with an angular velocity of about 5 radians per second.

## 3. The Magic Height

Daunted? don’t worry: the laws of mechanics can help. When you toss a pancake you swing the pan upwards for the toss. This swinging action imparts a natural angular velocity (spin) to the pancake. At the right height, the launch velocity and spin are perfectly matched. To work this out we need to know, R, the effective length of your arm that is swinging the frying pan. More about R later. The equation we can derive by comparing the spin of the pancake with the swing of the pan gives a very simple result:

If you know R, then the perfect toss formula gives you the height to aim for. Here are three examples.

The housewife’s toss: the upper arm hardly moves. The forearm swings upwards from the elbow holding the frying pan handle. The distance from the elbow to the centre of the pan is typically 60cm So R is 0.6m and the perfect toss formula gives a height of 0.5m (0.47 to be exact… but this really is not that exact)

##### For a forearm toss, aim for height of 0.5m

The Show off: This mammoth two armed throw, suitable for a real show off or someone wielding a cast iron pan. With arms outstretched swinging from the shoulder, R is effectively 0.9m and a height of 0.7 is the target.

##### For a two-armed toss, a height of 0.7m is needed … but watch the ceiling!

The Chef: This is a quick efficient flick of the wrist. Holding the pan near the rim and just flicking the wrist R is 20cm and the required height is just 16cm, just enough to clear the pan.

## 4. Aerodynamics

The previous analysis was simple mechanics, A-level maths and physics. The equations work anywhere, and would be particularly accurate on the moon with no air resistance. The magic heights would be just the same. Taking account of aerodynamics needs supercomputer power, but we can reach some broad conclusions: air both helps and hinders.

#### The landing

The pancake will tend to float down into the pan, rather like a leaf from a tree – only rather faster and heavier. It might seem more aerodynamic to drop down sideways into a heap on the pan, but air tends to have the opposite effect: try dropping a sheet of paper lying flat and repeat trying to drop it down vertically.

#### The take off

If the landing was helped by aerodynamics, the launch certainly is not! The pancake has air pressure holding it to the pan. Air pressure is 15 pounds per square inch, which is about ten thousand kilograms per square metre. On the pancake that amounts to a massive 700kg pressing down on the pancake, and the same amount up on the bottom of the pan. That air pressure holds the sucker on your satnav to the windscreen. The pancake could be acting like a giant sucker; the bigger it is the bigger the suction. Here are some helpful hints to lift off.

##### Make sure the pancake is not sticking to the pan, and can move freely from side to side

Did you need a physicist to tell you that?

The next trick is about how to break the air pressure. Lift-off can happen, and will happen, as air can creep in from the edges to fill the gap between the pancake and the pan. For bigger pancakes the area increases much faster than the amount of edge that the air has to creep in. Increasing a pancake from 10cm diameter to 30cm diameter, and the area goes up nine times while the perimeter increases by a factor of only three. So it gets three times harder to get air underneath the larger pancake. It does help to restrict the size of the pancake – imagine trying to toss a French crepe. The cook can also help:

##### Cook the pancake until it bubbles and blisters to roughen the bottom surface, before you toss it.

That allows the air to rapidly fill any void under the pancake and equalise the pressures.

As an alternative to a fast take off with air getting underneath the pancake, a skilful tosser may be able to slide the pancake out of the pan, without of course impairing the tossing and turning motion.