An explosion at ground level leaves a crater with a diameter that is proportional to the energy of the explosion raised to the 1/3 power; an explosion of 1 megaton of tnt leaves a crater with a 1 km diameter. an ancient impact crater is found with a 36 km diameter. what was the kinetic energy associated with that impact, in terms of (a) megatons of tnt (1 megaton yields 4.2 × 1015 j) and (b) hiroshima bomb equivalents (13 kilotons of tnt each)?

a)125 GT

b)9615384.6 Hiroshima bombs

Explanation:

We know that

D=k(E)^(1/3) (and I will keep D in km and E in MT of TNT)

D= diameter

Where k is a constant of proportionality.

We also have

1=k*(1)^(1/3), k=1

(a) For D=50, E=(D^3), so E= 125000 MT or 125 GT

(b) For Hiroshima equivalents, that's 125000 MT / 13 KT = 125000 MT / 0.013 MT = 9615384.6 Hiroshima bombs.

a) 3.024*10^12 MT

b) 2.326*10^14 units

Explanation:

Given:

- The relationship between diameter d of the crater and energy E is:

                                             d = k*E^(1/3)

- Where k is constant of proportionality

- Explosion of 1 MT of TNT creates 1 km diameter.

Find:

An ancient impact crater is found with a 36 km diameter

(a) megatons of TNT (1 megaton yields 4.2 × 1015 J)

(b) Hiroshima bomb equivalents (13 kilotons of TNT each)?

Solution:

- To evaluate constant k:

                                        1,000 = k*(4.2*10^15)^(1/3)

                                        k = 1.543*10^-5

- The relationship can now be expressed as:

                                       d = 1.543*10^-5*E^(1/3)

- For a crater of d = 36 km, the associated energy is:

                                      E = (36,000 / 1.543*10^-5)^3

                                      E = 1.2699*10^28 J

a)

                    1 MT of TNT   > 4.2 × 10^15 J

                                x         > 1.2699*10^28 J

                        x = 1.2699*10^28 /4.2 × 10^15  = 3.024*10^12 MT

b)     The E = 3.024*10^12 MT.

       1 unit of Hiroshima bomb is  = 0.013 MT of TNT

        Units of bomb = 3.024*10^12 MT / 0.013 MT = 2.326*10^14 units