Question

The energy yield of a nuclear weapon is often defined in terms of the equivalent mass of a conventional explosive. 1 ton of a conventional explosive releases 4.2 GJ. A typical nuclear warhead releases 250,000 times more, so the yield is expressed as 250 kilotons. That is a staggering explosion, but the asteroid impact that wiped out the dinosaurs was significantly greater. Assume that the asteroid was a sphere 10 km in diameter, with a density of 2500 kg/m3 and moving at 30 km/s. Part A: What energy was released at impact, in joules? Assume all kinetic energy of the asteroid to be released. Part B: What energy was released at impact, in kilotons? Assume all kinetic energy of the asteroid to be released.

Answer 1

A) $5.88\cdot 10^{23}J$

The kinetic energy of an object is given by

$K=\frac{1}{2}mv^2$

where

m is the mass of the object

v is the speed of the object

For the asteroid in the problem, we have

$d=2500 kg/m^3$ is the density

$r=\frac{10 km}{2}=5 km=5000 m$ is the radius

We can find its volume:

$V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (5000 m)^3=5.23\cdot 10^{11} m^3$

And so its mass

$m=d V =(2500 kg/m^3)(5.23\cdot 10^{11} m^3)=1.31\cdot 10^{15} kg$

while its speed is

$v=30 km/s=30,000 m/s$

So its kinetic energy is

$K=\frac{1}{2}(1.31\cdot 10^{15} kg)(30,000 m/s)^2=5.88\cdot 10^{23}J$

so, this is the energy released in the impact of the asteroid.

B)  $1.4\cdot 10^{11} kton$

The conversion factor is

$1 ton = 4.2 GJ = 4.2 \cdot 10^9 J\\1 kton = 4.2 \cdot 10^{12}J$

So the energy released in kilotons will be

$E=\frac{5.88\cdot 10^{23} J}{4.2\cdot 10^{12} J}=1.4\cdot 10^{11} kton$