A ball rolls off a flat roof and hits the ground 2 seconds later. The roof is ? meters high

The roof is 19.6 meters high.

Explanation:

As the ball rolls on the flat roof, when it starts falling down there is no initial velocity in the vertical axis, so we can define the initial vertical velocity as zero. (We define the time t= 0s as the moment when the ball starts falling down)

Now, the only force acting on the vertical axis will be the gravitational force, then the acceleration in the vertical axis is the vertical acceleration:

a(t) = -9.8m/s^2.

Now, to get the vertical velocity, we need to integrate over time, and get:

v(t) = (-9.8m/s^2)*t + v0

where v0 is the initial vertical velocity, and as we already know, this is zero.

v(t) = (-9.8m/s^2)*t

Now to get the position, we integrate again over time:

p(t) = (-4.9m/s^2)*t^2 + p0

Where p0 is the initial position, if we define the zero in the position as the ground, then p0 will be equal to the height of the roof.

p(t) = (-4.9m/s^2)*t^2 + h

And we know that the ball hits the ground at t = 2s, then we have:

p(2s) = 0m = (-4.9m/s^2)*(2s)^2 + h

           (4.9m/s^2)*(2s)^2 = h

                   19.6m = h

The height of the roof is 19.6 meters.

explanation:

newton's third law, for every action there is an equal and opposite reaction.

thermal energy is the answer