The only thing you can derive ...maybe ... from the given information is that
the stroller's x(t) is 1 + the mother's x(t). And if her progress is not strictly
along the x-axis, then even that statement isn't true.
If the stroller's v(t) and a(t) were not identical to the mother's v(t)and a(t), then
mother and stroller would "come apart". But you've told us that they're always
in contact, so their v(t) and a(t) must be identical.
A slightly more rigorous (and mathematical) argument is that since the mother's
and the stroller's x(t) differ (at most) only by a constant, then their derivatives
x'(t) (which is v(t)) and x''(t) (which is a(t)) are equal.
