A physics student fits a damper to a trolley of mass 2 kg that is joined to a fixed wall by a…

A physics student fits a damper to a trolley of mass $2\ \text{kg}$ that is joined to a fixed wall by a spring. The spring constant is $50\ \text{N/m}$, and the damper gives a force of $8\ \text{N s/m}$ times the velocity, against the motion. The student pulls the trolley $0.10\ \text{m}$ from rest and lets go. The motion obeys $2x'' + 8x' + 50x = 0$, where $x$ is the displacement in metres and $t$ is the time in seconds. She expects the trolley to swing to and fro with falling amplitude. She wants three numbers. The first is the angular frequency of the damped swing. The second is the time between two successive peaks. The third is the height of the first peak after the start as a fraction of the starting height $0.10\ \text{m}$. Give each answer to four decimal places.