Battleships! In 1905, as part of the Russo-Japanese War, the Imperial Japanese Navy (IJN) under the command of Admiral Togo engaged the Russian fleet under the command of Admiral Rozhestvensky in the Battle of Tsushima. This was the first major naval engagement to be dominated by "big guns." In this question we will model the 12" Armstrong Whitworth guns employed by the IJN. The guns are capable of firing a 390kg shell at a speed of 732ms-1. For simplicity, we shall assume that this shell is a sphere of diameter D such that the drag due to air resistance can be approximated as 1 FD CopAv² = 160D²y² = cv² = where Cp = 1/2 is the approximate drag coefficient for a sphere, A = Tt(D/2)² is the reference area, p z 1.225kgm-3 is the density of air at sea level, and v is the velocity, whence C = 16PD? The overall force experienced by our shell in flight is hence F = mg – cv²v, where m is the mass of the shell and g is the acceleration due to gravity, v= v and v^ = v/v, with v corresponding to the velocity vector. We shall take our coordinate system such that x^ points east, y^ points north, and z^ points up whence g--9.81z^. With this, let our gun be placed at the origin of our coordinate system, and assume our ship is at rest. We shall model the ship we are trying to hit with our shell as having a length of 121m, a beam of 23m, and being purely rectangular in shape. At t = 0 let the midpoint of this ship be located 4km to the north and 6km to the east of us, steaming with a speed of 7.5ms-1 in a south-southeast direction. (a) We are interested in approximating the position vector r(t) = [x(t), function of time. Given that v(t) = r'(t), a(t) = v'(t) = "r(t) and Newton's second law %3D of our shell as a F = ma, formulate a system of three second-order ordinary differential equations which describe the motion of our shell. (b) Rewrite this system as a set of six first-order ordinary differential equations.

Battleships! In 1905, as part of the Russo-Japanese War, the Imperial Japanese Navy (IJN) under the command of Admiral Togo engaged the Russian fleet under the command of Admiral Rozhestvensky in the Battle of Tsushima. This was the first major naval engagement to be dominated by "big guns." In this question we will model the 12" Armstrong Whitworth guns employed by the IJN. The guns are capable of firing a 390kg shell at a speed of 732ms-1. For simplicity, we shall assume that this shell is a sphere of diameter D such that the drag due to air resistance can be approximated as 1 FD CopAv² = 160D²y² = cv² = where Cp = 1/2 is the approximate drag coefficient for a sphere, A = Tt(D/2)² is the reference area, p z 1.225kgm-3 is the density of air at sea level, and v is the velocity, whence C = 16PD? The overall force experienced by our shell in flight is hence F = mg – cv²v, where m is the mass of the shell and g is the acceleration due to gravity, v= v and v^ = v/v, with v corresponding to the velocity vector. We shall take our coordinate system such that x^ points east, y^ points north, and z^ points up whence g--9.81z^. With this, let our gun be placed at the origin of our coordinate system, and assume our ship is at rest. We shall model the ship we are trying to hit with our shell as having a length of 121m, a beam of 23m, and being purely rectangular in shape. At t = 0 let the midpoint of this ship be located 4km to the north and 6km to the east of us, steaming with a speed of 7.5ms-1 in a south-southeast direction. (a) We are interested in approximating the position vector r(t) = [x(t), function of time. Given that v(t) = r'(t), a(t) = v'(t) = "r(t) and Newton's second law %3D of our shell as a F = ma, formulate a system of three second-order ordinary differential equations which describe the motion of our shell. (b) Rewrite this system as a set of six first-order ordinary differential equations.

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Battleships! In 1905, as part of the Russo-Japanese War, the Imperial Japanese Navy (IJN) under the
command of Admiral Togo engaged the Russian fleet under the command of Admiral Rozhestvensky
in the Battle of Tsushima. This was the first major naval engagement to be dominated by "big guns."
In this question we will model the 12" Armstrong Whitworth guns employed by the IJN.
The guns are capable of firing a 390kg shell at a speed of 732ms-1. For simplicity, we shall assume
that this shell is a sphere of diameter D such that the drag due to air resistance can be
approximated as
1
FD
2
CppAv²:
= cv?
16
where Cp= 1/2 is the approximate drag coefficient for a sphere, A = T(D/2)2 is the reference area, p
- 1.225kgm-3 is the density of air at sea level, and v is the velocity, whence
C =
16PD?
The overall force experienced by our shell in flight is hence
F = mg -
- cv³v ,
where m is the mass of the shell and g is the acceleration due to gravity, v= v and v^ = v/v, with
v corresponding to the velocity vector.
We shall take our coordinate system such that x^ points east, y^ points north, and z^ points up
whence g-9.81z^. With this, let our gun be placed at the origin of our coordinate system, and
assume our ship is at rest. We shall model the ship we are trying to hit with our shell as having a
length of 121m, a beam of 23m, and being purely rectangular in shape. At t= 0 let the midpoint of
this ship be located 4km to the north and 6km to the east of us, steaming with a speed of 7.5ms-1 in
a south-southeast direction.
(a) We are interested in approximating the position vector r(t)
[x(t),y(t),z(t)] of our shell as a
%3D
function of time. Given that v(t) = r'(t), a(t) = v'(t) = "r(t) and Newton's second law
F = ma,
formulate a system of three second-order ordinary differential equations which describe the
motion of our shell.
(b) Rewrite this system as a set of six first-order ordinary differential equations.

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