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.

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.