# How can the SUVAT equations be derived from Newton’s laws of motion?

I would argue that the SUVAT equations, which correctly describe all constant-acceleration motion (including hypothetical situations governed by unusual laws of dynamics), cannot possibly be derived from a specific set of laws of dynamics, such as Newton’s laws of motion.

Furthermore, the derivation of the SUVAT equations is well-established in, for example, How are the SUVAT equations derived? and Does the SUVAT equations of motion (Kinematics) come from some differential equation?. These derivations have nothing to do with any laws of dynamics.

Yet, the top-voted answer to an active SUVAT-themed question says:

They are in fact three results derived from the distillation of Newton's Laws: $$\mathbf f = \dfrac {\mathrm d} {\mathrm d t} (m \mathbf v)$$

which differential equation is solved, where $$\mathbf f$$ is set to a constant (and $$m$$ is taken for granted as being constant also).

(that question was based on three of the SUVAT equations)

When I challenged this statement, using similar arguments to the ones above, and asked for more details, I did not receive any responses that helped me. I did receive this response:

As for it "making no sense" to derive these equations of motion from Newton's equations, that is exactly the way this was introduced in one of my applied maths courses somewhere during my education -- as a way of unifying the SUVAT equations with the facts of reality.

Another user, not involved in that chat, tagged the question .

So the idea that the SUVAT equations can be derived from Newton’s laws of motion seems to be well-established. Where did I go wrong, claiming that such a derivation is impossible, and what is the derivation?

## 3 Answers

Arguably the suvat equations are more mathematical than physical. We start with axioms – the definitions of velocity in terms of displacement and time, and acceleration in terms of velocity and time – and derive the equations. Indeed I believe that Galileo himself thought of kinematics as part of geometry. And this was more than 250 years before Minkowski!

It's possible, I suppose, that a teacher might motivate the study of constant acceleration by considering a body acted on by a constant force, such as a trolley pulled by a weight connected to the trolley by a string passing over a pulley, but I'd hope that the teacher would point out later that no specific set-up is required by the equations themselves.

The constant-acceleration kinematic equations are most useful as a special case of Newton's Second Law. In other words, it is incredibly common to look at situations where the forces is constant, and in such cases the solution to Newton's Second Law (i.e., how $$s$$, $$v$$, etc. behave as functions of time) is given by the SUVAT equations.

To be explicit about this, if $$\mathbf{f}$$ and $$m$$ are constants (as is typically the case in introductory mechanics problems), then we have for 1-D motion $$f = m \frac{dv}{dt} \quad \Rightarrow \quad a = \frac{f}{m} = \text{const.}$$ and from this point on the derivations in the threads you've linked follow.

If I understand your objection correctly, it's that we could imagine some parallel universe where the laws of dynamics were different. For example, maybe we would have $$f^2 = d(mv)/dt$$. In this universe, for constant force and constant mass, we would still have constant acceleration, and the SUVAT equations would still hold.

And I suppose that's true, but it doesn't mean that the SUVAT equations can't be derived from Newton's Second Law under certain assumptions; it just means that there are other sets of assumptions that would lead to the same result. To put it another way, Newton's Second Law (with constant force and mass) implies the SUVAT equations; but knowing that the SUVAT equations hold in all cases for constant force and mass does not imply Newton's Second Law.

The SUVAT equations are simply a mathematical consequence of constant acceleration

$$\displaystyle \frac {d^2x}{dt^2} = \frac {dv}{dt} = a$$

together with the initial conditions $$x(0) = 0$$ and $$v(0) = u$$. There is no mention here of force, so they have no logical connection with Newton's Laws. If an object moved with constant acceleration because it was being propelled by pink unicorns rather than following Newton's Laws, the SUVAT equations would still apply to it.