I'm not satisfied with the argument that this satisfies Newton's first law. The fact that it has zero acceleration before T and at the moment T is necessary but it isn't sufficient.
I take issue with the interpretation of Newton's first law. What's happening here is some sleight of hand that allows the author to interpret Newton's first law in such a way that it becomes completely empty - it simply becomes a restatement of the second law. Indeed, we look at it in its instantaneous form, which means that what we actually confirm is that dv/dt=k(t)*F(t), where F(t) is force at any time T and k(T) is a nonzero function - this is the functional equivalent of the statement that "In the absence of a net external force, a body is unaccelerated.". This is of course just a more general statement for F=ma where k(T) = 1/m. So that leads me to conclude that either this instantaneous version of the law isn't sufficient, or that it's a misinterpretation of the law, or that Newton is an idiot and didn't realize that his first law is redundant.
Personally I just think that this is a misreading of the first law - I read it as saying that force is causative to changes in velocity, and not the reverse, and hence that the logic in this argument suddenly stops making sense, and only perpetual rest is a correct solution, since we cannot resolve the trajectory from the forces anymore, we have to presuppose the trajectory before finding implied forces, thus breaking the causative relationship implied by the first law. I don't think that when Newton wrote that rest in the inertial frame is something that must be broken by action, he meant something that is straightforwardly implied by the second law, as of course he knew the elementary calculus necessary to see that a weaker statement is redundant.
I take issue with the interpretation of Newton's first law. What's happening here is some sleight of hand that allows the author to interpret Newton's first law in such a way that it becomes completely empty - it simply becomes a restatement of the second law. Indeed, we look at it in its instantaneous form, which means that what we actually confirm is that dv/dt=k(t)*F(t), where F(t) is force at any time T and k(T) is a nonzero function - this is the functional equivalent of the statement that "In the absence of a net external force, a body is unaccelerated.". This is of course just a more general statement for F=ma where k(T) = 1/m. So that leads me to conclude that either this instantaneous version of the law isn't sufficient, or that it's a misinterpretation of the law, or that Newton is an idiot and didn't realize that his first law is redundant.
Personally I just think that this is a misreading of the first law - I read it as saying that force is causative to changes in velocity, and not the reverse, and hence that the logic in this argument suddenly stops making sense, and only perpetual rest is a correct solution, since we cannot resolve the trajectory from the forces anymore, we have to presuppose the trajectory before finding implied forces, thus breaking the causative relationship implied by the first law. I don't think that when Newton wrote that rest in the inertial frame is something that must be broken by action, he meant something that is straightforwardly implied by the second law, as of course he knew the elementary calculus necessary to see that a weaker statement is redundant.