Problem 1C-10*: Hyperbolic functions and hyperbolic orbit
Hyperbolic functions sinh(z) and cosh(z) are defined as
(a) Prove sinh(-z)=-sinh(z) and cosh(-z)=cosh(z).
(b) Prove cosh2(z)-sinh2(z) = 1.
(c) Prove sinh(A+B) = sinh(A)·cosh(B)+cosh(A)·sinh(B).
(d) Prove cosh(A+B) = cosh(A)·cosh(B)+sinh(A)·sinh(B).
(e) If y(t) = A·cosh(ωt+α) and x(t) = B·sinh(ωt+β), what is the trajectory of the motion?
(f) Calculate the velocity and the acceleration vectors for this motion.
so the trajectory is a hyperbola.
(f) Let r(t) = (x(t), y(t)) = (B·sinh(ωt+α), A·cosh(ωt+α)). Then
This means that the particle moves faster and faster as t increases and the particle moves away from the origin. For the acceleration vector we have
so |a(t)| also increases to infinity as t → +∞. It is also interesting to note that
so a(t) and r(t) lie on the same line and are pointing at the same direction.
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