Given
A ball is rolling down a hill on the earth without any friction and starts with a velocity of 0 m/s at the peak and ends up with 30 m/s at the valley.
Find
Solution
\(
\begin{array}{l}
E_{pot} + E_{kin} = const \\
E_{pot} = m \cdot g \cdot h \\
E_{kin} = \frac{m \cdot v^2}{2} \\\\
\end{array}
\)
\(
\begin{align*}
E_{peak} &= E_{valley} \\
E_{pot_{peak}} + E_{kin_{peak}} &= E_{pot_{valley}} + E_{kin_{valley}} \\
E_{pot}(h=h_{peak}) + E_{kin}(v=0\frac{m}{s}) &= E_{pot}(h=0m) + E_{kin}(v=30\frac{m}{s}) \\
E_{pot}(h=h_{peak}) &= E_{kin}(v=30\frac{m}{s}) \\
m \cdot g \cdot h_{peak} &= \frac{m \cdot (30\frac{m}{s})^2}{2} \\
h_{peak} &= \frac{(30\frac{m}{s})^2}{2 \cdot g} \\
h_{peak} &= \frac{(30\frac{m}{s})^2}{2 \cdot 9,81\frac{m}{s^2}} \\
h_{peak} &\approx 45,87m
\end{align*}
\)
Answer