# Is the Coriolis acceleration formula just an approximation?

The Coriolis acceleration given in textbooks is:

$a_{cor} = -2\omega v$

When trying to understand the derivation, it seems as though this is just an approximation. And to illustrate this I give a simple example of a rotating platform with two people standing on it. Person A stands at the axis, so therefore will have no centripetal acceleration. Person B stands at the far edge of the platform, and they both face each other. A throws a ball to B at t=0. To keep the calculations simple, the angular velocity of the platform is 1 rad/s, the radius of the platform is 1m, and the velocity at which the ball is thrown is 1m/s.

After one second, the ball will travel radially outwards one meter, meaning that its position is now the same as B at t=0. B will now have moved an angular displacement of 1 rad to position B'.

According to the formula, $a_{cor} = -2ms^{-2}$. From this I integrate twice to find the displacement of the ball relative to B after this one second, so:

$s = -\frac 1 2 a_{cor}t^2 =$ -1m.

This doesn't seem accurate to me. B has travelled 1m from its original position, so I guess it's correct if you're measuring the distance along the circumference. But this is the arc length and not the actual displacement of the ball, which would be the chord length from B to B'. This is slightly less than 1 meter.

Am I right or am I not understanding this correctly?

• The $t^2$ formula for distance as a function of acceleration and time is only valid if the acceleration is constant. If the acceleration depends on the position (as in this case), then one has to solve a differential equation to get the correct trajectory. The formula is correct (because the quantities are all differential) but application of the simple $t^2$ formula is not. – CuriousOne Dec 21 '15 at 1:51
• @CuriousOne: But in this case the coriolis force is constant, since all the ball velocity, the rotation speed and the angle between those is constant. – rodrigo Dec 21 '15 at 2:00
• @rodrigo: The force is not constant, only its magnitude is. – CuriousOne Dec 21 '15 at 2:05