What is a transverse wave? how do the particles in the medium move in relation to the energy of the wave? does this wave require a medium?

A transverse wave is a wave where the particles in the medium move perpendicular (at right angles) to the direction of the source or its propagation (think of a snake slithering through grass) an example of a transverse wave could be a light wave. Light waves for instance don’t need a medium in order to propagate but transverse waves in general do need a medium.

Kinetic energy = 1/2mv^2. since the mass is constant we can form a ratio of it's kinetic energy at 60 divided by that at 30 to see ke(60)/ke(30) = +(60^2)/30^2). this may be said to 300% increase.