physics-online.info: Problem 1C-10* - Hyperbolic functions and hyperbolic orbit

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.

(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.

Solution:

(a)

(b)

(c)

(d)

(e)

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.