How do you calculate the power of an object without finding time? given only mass, height and speed. : ))

Well, first of all, an 'object' doesn't have power.  Power is the rate at which work is done or energy is moved or transferred.  An object can have energy, and can carry energy from place to place, but you wouldn't ever say that the object 'has' power.  

In order to calculate power, you need something like force, distance and time, or work and time, or energy and time.  There's really no such thing as "the power of an object".

The only other remark I can offer is that if you do have mass, height, and a speed given, then you actually do have 'time'.  That speed that you have is distance divided by time.  So time is actually buried there in the data that you have.

what experimental evidence led scientists to change from the previous model to this one?

a few of the positive particles aimed at a gold foil seemed to bounce back.

the colors of light emitted from heated atoms had very specific energies.

experiments with water vapor showed that elements combine in specific proportions.

cathode rays were bent in the same way whenever a magnet was brought near them.