A particle moves along a straight track so that its velocity v is a continuous function of time t(s). The position function is denoted s(t)(m). Throughout this problem, assume all required derivatives exist and are continuous where needed.
(a) Derive an expression for s(t) in terms of v and a single constant using an arbitrary reference time t0. Clearly state the role of the constant.
(b) Suppose s(3)=7 and the explicit form of v is not known. Rewrite your result from part (a) so that it yields s(t) for any t without the unknown constant by choosing an appropriate reference time.
(c) Explain why the value of ∫36v(u)du may be negative even though the particle’s speed is always non-negative, and interpret the physical meaning of a negative value in this context.