Sparks occur when the electric field in air exceeds 3 x 106 N/C. This is because free electrons normally present in air are accelerated to such high speeds that their kinetic energy will overcome the potential energy holding other electrons to atoms. When those electrons rearrange themselves after such a collision, a flash of light is emitted. Let us suppose that the work done on an electron must give it an energy of 3 x 10-19 J to cause this ionization. How far does an electron involved in making in a spark travel through the air before it collides with an atom

h = 5.38 10¹⁶ m

Explanation:

Let's start this exercise by assuming that all the potential energy of the electron is converted into kinetic energy, let's use the conservation of energy

starting point. Just before ionization

          Em₀ = U = qE

final point. Right after ionization

           Em_f = K = ½ m v²

Energy is conserved

           Em₀ = Em_f

           q E = ½ m v²

           v² = 2qE / m

Now we can use the relationship between net work and kinetic energy

           W_net = ΔK

net work is the work done by the electron minus the binding energy with the atom, called the work function, Ф = 3 10-19 J

           W - Ф = K_f - K₀

we assume that the electron converts all its initial initial kinetic energy to be zero

           W -Ф = ½ m v² - 0

            W = ½ m v² +Ф

we substitute

           W = 1/2 m 2qE/m + E

           W =  qE +Ф

           W = 1.6 10⁻¹⁹ 3 10⁶ + 3 10⁻¹⁹

            W = 4.8 10⁻¹³ + 3 10⁻¹⁹

           W = 4.8 10⁻¹³ J

When the electron is in air, its kinetic energy can be transformed into gravitational potential energy

As the electron is in the air, all work is transformed into scientific energy

           W = K

starting point Em₀ = K = W

end point Em_F = U = m g h

energy conservation Em₀ = Em_f

                       W = m g h

                       h =

let's calculate

                       h =

                       h = 5.38 10¹⁶ m

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .