fluid dynamics - Can cannonballs go through water? - Physics

itemprop="text">

In the recent Spielberg/Jackson Tintin
movie, there is a scene where Red Rackham and Captain Haddock's ships are fighting, and
cannons are fired. The cannonball is shown at one point to go through a wave, and
inflict serious damage on the other ship. I know that bullets stop in water; do
cannonballs, with their greater weight, continue with enough force to inflict damage?

style="font-weight: bold;">

Answer

What distance can a cannonball traverse thru
water without losing too much kinetic energy? For a back-of-the-envelope calculation we
start from the observation that this distance scales with the ratio of the kinetic
energy of the cannonball and the drag force exerted on the cannonball.

Let's denote the ball's radius by $R$, its
speed by $v$, and its mass density by $\rho_{ball}$. The kinetic energy $E_k$ equals
$\frac 1 2 M v^2 = \frac{2 \pi}{3} \rho_{ball} R^3 v^2$.

The drag force $F_d$ is given by
$\frac 1 2 C_d \rho_{water} v^2 A = \frac {\pi}{2} C_d \rho_{water} v^2 R^2$. Here,
$C_d$ denotes the drag coefficient for a sphere.

The maximum distance $L _{max}$ that can be
traversed by a cannonball $L_{max} = E_k/F_d$ is therefore $\frac 4 3 \frac {R}{C_d}
\frac {\rho_{ball}}{\rho_{water}}$. For typical values (
$\frac{\rho_{ball}}{\rho_{water}} < 8$ and $C_d > 0.1$, see href="http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html">here), we
find $L_{max} < 100 R$.

In other words, a
cannonball loses much of its kinetic energy when it traverses a layer of water larger
than about fifty times its diameter.