Question

Give a step by step description of the physics and mathematics needed to calculate the trajectory of a water bottle rocket in three dimensions. In addition, if the location of the landing of a water bottle rocket could be controlled, what is the physics and mathematics needed to establish control? Consider the question in three dimensions.

Answer 1

Please consider the ans-

A water rocket is basically an upside down fizzy drinks bottle, which has had a ‘nose’ cone and some fins added. The body of the rocket is filled with water to some desired amount, typically about 40% of the volume. The rocket is then mounted on a launch tube which is quite similar to that used by a compressed air rocket. Air is pumped into the bottle rocket to pressurize the bottle and thrust is generated when the water is expelled from the rocket through the nozzle at the bottom. The weight of the bottle rocket is constantly changing during the powered ascent, because the water is leaving the rocket. As the water leaves the rocket, the volume occupied by the pressurized air increases. The increasing air volume decreases the pressure of the air, which decreases the mass flow rate of water through the nozzle, and decreases the amount of thrust being produced. Weight and thrust are constantly changing during the powered portion of the flight. When all of the water has been expelled, there may be a difference in pressure between the air inside the bottle and the external, free stream pressure. The difference in pressure produces an additional small amount of thrust as the pressure inside the bottle decreases to ambient pressure. When the pressures equalize, there is no longer any thrust produced by the rocket, and the rocket begins a coasting ascent.

Dynamics of the water bottle rocket--

The first force is gravity, and its magnitude is simply mg, where m is the mass of the rocket. However, remember that the rocket mass will change during the flight as the rocket expels its water. The next force is the thrust, and we will say some more about how this is calculated below. For now we note that we assume the thrust acts along the axis of the rocket. And the last force is the aerodynamic drag which has the form:

F(drag) = kv^2

where v is the air speed, and k is a constant which depends on how aerodynamic your rocket is. We assume that the rocket is aerodynamically stable and that the drag force always acts in the opposite direction to the velocity vector. To simulate the dynamics of the rocket we calculate the sum of the three forces mentioned above, and resolve the force into its x and y components. We then use Newton’s third law (F = ma) to calculate acceleration due to the net force on the rocket. At this point we make an approximation:

we estimate the change in each component of velocity due to each component of the acceleration a as: ∆ v(x) ≈ a(x)∆ t

which is strictly only accurate in the limit ∆ t →0. Having an estimate for the change in velocity during the time ∆t, we can then estimate the change in each component of the position of the rocket as:

∆ x ≈ v(x)∆ t This process is repeated typically a few hundred or a few thousand times during the flight with short time steps, and seems to give a fair approximation to realistic flight dynamics.

The trajectories are governed by Ordinary Differential Equations (ODEs) which give the time rate of change of each state variable. These are obtained from the definition of velocity,from Newton’s 2nd Law, and from mass conservation.h˙ = V

V˙ = F/m

m˙ = −m˙ (fuel)

The total force F on the rocket has three contributions: the gravity force, the aerodynamicdrag force, and the thrust.

F =−mg − D + T , if V > 0

      −mg + D − T , if V < 0

The rocket’s thrust to the propellant mass flow rate m˙ fuel is via the exhaust velocity (ue).T = m˙ (fuel) ue

A convenient way to express the drag is

D = 1/2*ρ(V^2) CD(A)

Where t time, h altitude, V velocity, positive upwards, F total force, positive upwards, D aerodynamic drag, T propulsive thrust, ∆t time step, ρ air density, g gravitational acceleration, m mass, CD drag coefficient, A drag reference area, m˙ fuel fuel mass flow rate, ue exhaust velocity.

Assuming a flat Earth with a uniform gravity field, and no air resistance, a projectile launched with specific initial conditions will have a predictable range.

The following applies for ranges which are small compared to the size of the Earth.

where

d is the total horizontal distance travelled by the projectile.

v is the velocity at which the projectile is launched

g is the gravitational acceleration—usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface

θ is the angle at which the projectile is launched