I will assume you're talking about the tangential speed of a particle following a circular trajectory. But you can also give sense to the tangential speed of arbitrary trajectories by the use of derivatives and tangent vectors. A formula for the tangential speed, for a circular motion, can be constructed...

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I will assume you're talking about the tangential speed of a particle following a circular trajectory. But you can also give sense to the tangential speed of arbitrary trajectories by the use of derivatives and tangent vectors.

A formula for the tangential speed, for a circular motion, can be constructed based on the data available to you. For instance, if it the centripetal force `F_{cp} ` acting on the particle is given to you, then you can find the tangential speed of the trajectory as a function of the radius of the trajectory and the mass of the particle.

`v = sqrt({F_{cp}R}/{m})`

In this answer, I will assume the values of the radius of the trajectory and the angular speed (that is, the number of revolutions per unit time made by the particle) are given to you. So, for a circular motion of constant angular speed `w` and constant radius `R` , the tangential speed is given by the following formula:

`v=wR`

But remember, this is only one of the many ways of obtaining the tangential speed of a circular trajectory. It all depends on the context and values given to you. As an example, there are problems in which the angular speed is not constant with time, and this changes the formula.

If the angular speed is not constant, and the angular acceleration `alpha` and the radius `R` are constant with time, then the tangential speed will be given by the following formula:

`v(t)=(w_0 + alpha t)R` ` `

Notice that the tangent speed is now a function of time.

In general, you should analyze your problem and construct a formula for calculating the tangential speed, or any other variable. Try to avoid these specific formulas for specific cases and focus on the construction of them.