## 3 Analytical and Numerical Solutions
### Case of Constant Density and Incompressible Hull
It was assumed above that the buoyancy change takes place instantaneously. In reality the buoyancy change occurs during a finite interval of time \(t_{0}\). Some results of numerical solution of the acceleration problem (5) for the cylinder on assumption of linear law of the buoyancy change are presented in Figures 5 and 6. Numerically found solutions of the Equations (7) and (8) are plotted in Figures 5 and 6 with dashed lines. Depending of a given depth corridor \(H\) and with account of the body inertia, it is necessary to switch the buoyancy engine to ascension at a certain moment of time \(t_{\text{as}}\). Therewith starts the process of deceleration of the vehicle until it stops completely. \[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]0 Speed of submersion of a sphere versus time in accelerated motion for an instantaneous change in buoyancy (\(t_{0}=0\)) for various \(\eta\), ds = 0.25 m.  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]1 Submersion of a sphere versus time in accelerated motion for an instantaneous change in buoyancy (\(t_{0}=0\)) and various \(\eta\), ds = 0.25 m.  
Consider first the analytical description of the deceleration process for the case of an instantaneous (\(t_{0}=0\)) relative buoyancy change from \(-\eta\) to \(\eta\). Accordingly, the problem acquires the form  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]2  
Note that in Equation (12), the time is counted from the moment of the start of deceleration. Integrating Equation (12) with initial conditions, we obtain the following expressions for instantaneous speed of deceleration and the distance covered from the start of deceleration until the vehicle completely stops in the form  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]3  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]4 Submersion of the cylinder versus time in accelerated motion for various magnitudes of \(t_{0}\), \(\eta=0.01\), \(l_{c}=1\) m.  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]5 Speed of submersion of the cylinder versus time in accelerated motion and various magnitudes of \(t_{0}\), \(\eta=0.01\), \(l_{c}=1\) m.  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]6  
As seen from Equation (13), the vehicle completely stops at the moment of time  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]7  
Substituting Equation (15) into Equation (14), we obtain the deceleration distance  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]8  
It follows from Equation (16) that the deceleration distance as well as acceleration distance--see Equation (11)--do not depend on the relative volume of replacement. After the time moment \(t=t_{d}\), the vehicle starts to ascend. For a description of the acceleration in ascending motion, it is necessary to consider the corresponding equation  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]9  
with initial conditions \(z(t_{d})=z_{md}\), \(\dot{z}(t_{d})=0\).