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**Wave Motion on a Hanging Rope**

Consider a rope of mass \( m \) and length \( L \) suspended from a ceiling.

1. **Wave Speed on the Rope**: Demonstrate that the wave speed at a distance \( y \) above the lower end of the rope is given by the formula:
\[
v = \sqrt{gy}
\]
where \( g \) is the acceleration due to gravity.

2. **Pulse Travel Time**: Show that the time \( \Delta t \) for a pulse to travel the full length of the rope is:
\[
\Delta t = 2\sqrt{\frac{L}{g}}
\]

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