Domain A.2: Newtonian Mechanics
A foundation for particle motion, multi-particle systems, gravitation, and the rocket equation.
1. Newton’s Laws of Motion
1.1 First Law — Inertial Motion
In an inertial frame, a particle maintains constant velocity unless a net external force acts on it. If the resultant force is zero,
\[ \mathbf{F}_{\text{net}} = \mathbf{0}, \qquad\Rightarrow\qquad \mathbf{v} = \text{constant},\; \mathbf{a} = \mathbf{0}. \]
1.2 Second Law — Dynamics
For a particle of mass \(m\) and velocity \(\mathbf{v}\), the net force equals the inertial time derivative of the linear momentum:
\[ \mathbf{F}_{\text{net}} = \left(\frac{d}{dt}\right)_N (m\mathbf{v}) = \left(\frac{d\mathbf{p}}{dt}\right)_N, \qquad \mathbf{p} = m\mathbf{v}. \]
If the mass is constant in time, this reduces to the familiar form
\[ \mathbf{F}_{\text{net}} = m\mathbf{a}. \]
All derivatives here are taken with respect to an inertial reference frame \(N\).
1.3 Third Law — Action and Reaction
If body 1 exerts a force on body 2, then body 2 exerts an equal and opposite force on body 1:
\[ \mathbf{F}_{12} = -\,\mathbf{F}_{21}. \]
These forces are collinear and act on different bodies.
2. Universal Gravitation
Consider two point masses \(m_1\) and \(m_2\) with position vectors \(\mathbf{r}_1\) and \(\mathbf{r}_2\). Define
\[ \mathbf{r}_{12} = \mathbf{r}_2 - \mathbf{r}_1, \qquad r_{12} = \|\mathbf{r}_{12}\|. \]
The gravitational force on \(m_2\) due to \(m_1\) is
\[ \mathbf{F}_{12} = -\,G\,\frac{m_1 m_2}{r_{12}^3}\,\mathbf{r}_{12}, \]
where \(G\) is the gravitational constant. Gravitation is a conservative force, so there exists a potential \(V_g(\mathbf{r})\) such that
\[ \mathbf{F}_g = -\nabla V_g. \]
3. Dynamics of a Single Particle
We now look at how a single particle moves under applied forces, how its energy changes, and how momentum appears in the description.
3.1 Equation of Motion
For a particle of mass \(m\), Newton’s second law gives
\[ m\mathbf{a} = \mathbf{F}_{\text{net}}, \qquad \mathbf{a} = \left(\frac{d\mathbf{v}}{dt}\right)_N. \]
Once the net force is known as a function of time, position, or velocity, the trajectory \(\mathbf{r}(t)\) follows from integrating this equation.
3.2 Motion Under a Constant Force
If the net force \(\mathbf{F}\) is constant, the acceleration is constant:
\[ \mathbf{a} = \frac{\mathbf{F}}{m}. \]
Integrating once gives the velocity:
\[ \mathbf{v}(t) = \mathbf{v}_0 + \mathbf{a}(t - t_0), \]
and integrating again yields the position:
\[ \mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 (t - t_0) + \tfrac{1}{2}\,\mathbf{a}(t - t_0)^2. \]
Typical examples include projectile motion near Earth in uniform gravity and simple constant-thrust segments in trajectory design.
3.3 Motion Under a Variable Force
If the net force depends on time, position, or velocity:
\[ m\mathbf{a}(t) = \mathbf{F}(\mathbf{r},\mathbf{v},t), \]
we obtain a differential equation that must be solved either analytically (for special force models) or numerically (e.g., Runge–Kutta schemes, symplectic integrators). This is the basis of orbital propagation with gravity and drag, re-entry modeling, and nonlinear attitude dynamics.
3.4 Work, Energy, and Power
The instantaneous power delivered by a force is
\[ P = \mathbf{F}\cdot\mathbf{v}. \]
The work done along a path \(C\) is
\[ W = \int_C \mathbf{F}\cdot d\mathbf{r}. \]
The kinetic energy of a particle is
\[ T = \tfrac{1}{2} m v^2. \]
From Newton’s law, one can show the work–energy theorem:
\[ W_{1\to2} = T_2 - T_1. \]
Thus, regardless of the force complexity, the net work equals the change in kinetic energy.
3.5 Conservative Forces and Potential Energy
A force is conservative if the work between two points depends only on the endpoints:
\[ W_{1\to2} = -\big[ V(\mathbf{r}_2) - V(\mathbf{r}_1) \big]. \]
In that case,
\[ \mathbf{F} = -\nabla V(\mathbf{r}), \]
where \(V(\mathbf{r})\) is the potential energy. For motion under conservative forces, the total mechanical energy
\[ E = T + V \]
remains constant: \(E = \text{constant}\). Energy conservation is central in orbital mechanics and spacecraft motion.
3.6 Linear Momentum
The linear momentum of a particle is
\[ \mathbf{p} = m\mathbf{v}. \]
Newton’s law can be written as
\[ \mathbf{F}_{\text{net}} = \left(\frac{d\mathbf{p}}{dt}\right)_N. \]
If the net external force vanishes,
\[ \mathbf{F}_{\text{net}} = \mathbf{0} \;\Rightarrow\; \mathbf{p} = \text{constant}, \]
so momentum is conserved and the particle moves with constant velocity.
3.7 Angular Momentum About a Point
Let \(P\) be a reference point with position \(\mathbf{r}_P\). The particle position relative to \(P\) is
\[ \mathbf{r}_{/P} = \mathbf{r} - \mathbf{r}_P. \]
The angular momentum about \(P\) is
\[ \mathbf{H}_P = \mathbf{r}_{/P} \times \mathbf{p}, \]
and if \(\mathbf{L}_P\) is the torque about \(P\),
\[ \left(\frac{d\mathbf{H}_P}{dt}\right)_N = \mathbf{L}_P. \]
Angular momentum and its conservation are key in orbital mechanics (Kepler’s second law) and in attitude dynamics.
4. Systems of Particles
A system of particles can represent a spacecraft, aircraft, or any multi-body structure before we idealize it as a rigid body. The goal is to describe overall motion in terms of the center of mass and internal motion.
4.1 Center of Mass
For particles \(i = 1,\dots,N\) with masses \(m_i\) and position vectors \(\mathbf{r}_i\):
\[ M = \sum_{i=1}^N m_i, \qquad \mathbf{R}_c = \frac{1}{M}\sum_{i=1}^N m_i \mathbf{r}_i. \]
The center-of-mass acceleration is
\[ \mathbf{a}_c = \left(\frac{d^2\mathbf{R}_c}{dt^2}\right)_N. \]
If \(\mathbf{F}_{\text{ext}}\) is the sum of all external forces on the system,
\[ \mathbf{F}_{\text{ext}} = M\mathbf{a}_c. \]
Thus, the entire system moves as if all mass were concentrated at the COM and acted on by the net external force (the “super-particle” idea).
4.2 Kinetic Energy Decomposition
The total kinetic energy is
\[ T = \sum_{i=1}^N \frac{1}{2} m_i v_i^2 = \frac{1}{2} M V_c^2 + T_{\text{rel}}, \]
where \(\tfrac{1}{2}MV_c^2\) is the kinetic energy of COM translation and \(T_{\text{rel}}\) is the kinetic energy due to motion relative to the COM (vibration, rotation, deformation). This decomposition is the starting point for rigid-body energy expressions.
4.3 Total Linear Momentum and Collisions
The total linear momentum of the system is
\[ \mathbf{p} = \sum_{i=1}^N m_i \mathbf{v}_i = M\mathbf{V}_c. \]
If there is no external force,
\[ \mathbf{F}_{\text{ext}} = \mathbf{0} \quad\Rightarrow\quad \mathbf{p} = \text{constant}. \]
This remains true even during collisions, because collision forces are internal and cancel in pairs, and the net external force is negligible over the short collision time.
However, kinetic energy may not be conserved:
- Elastic collision: momentum and kinetic energy conserved.
- Inelastic collision: momentum conserved, kinetic energy decreases.
- Perfectly inelastic: bodies stick together after impact.
Example in 1D:
\[ m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'. \]
Momentum conservation is a primary tool for analyzing impacts, docking, landing events, and multi-body interactions in aerospace systems.
4.4 System Angular Momentum
The angular momentum of an \(N\)-particle system about point \(P\) is
\[ \mathbf{H}_P = \sum_{i=1}^N \mathbf{r}_{i/P} \times m_i \mathbf{v}_i, \]
and the rate of change is governed by the external torque:
\[ \left(\frac{d\mathbf{H}_P}{dt}\right)_N = \mathbf{L}_P, \]
where \(\mathbf{L}_P\) is the sum of all external moments about \(P\). Internal forces form action–reaction pairs and give zero net torque.
4.5 Decomposition: COM Motion, Rigid Rotation, and Deformation
Let \(P\) be the inertial origin \(O\), so \(\mathbf{r}_{i/P} = \mathbf{r}_{i/O}\) and \(\mathbf{R}_P = \mathbf{0}\). For a deformable body or a cluster of particles attached to a nominal rigid body, each particle velocity can be written as
\[ \mathbf{v}_i = \mathbf{V}_c + \boldsymbol{\omega} \times \mathbf{r}_i' + \mathbf{v}_i', \]
where \(\mathbf{V}_c\) is the COM velocity, \(\boldsymbol{\omega}\) is the angular velocity, \(\mathbf{r}_i'\) is the position relative to the COM, and \(\mathbf{v}_i'\) is the internal (deformation) velocity.
Substituting into the angular momentum expression gives
\[ \mathbf{H}_O = M\mathbf{R}_c \times \mathbf{V}_c + \mathbf{I}_c \boldsymbol{\omega} + \sum \mathbf{r}_i' \times m_i \mathbf{v}_i'. \]
The three terms represent COM translation, rigid-body rotation, and deformation effects. If internal velocities satisfy \(\mathbf{v}_i' = \mathbf{0}\) or are parallel to \(\mathbf{r}_i'\), the deformation term vanishes and the body behaves like a rigid body with
\[ \mathbf{H}_O = \mathbf{I}_c \boldsymbol{\omega}, \]
the standard rigid-body angular momentum relation used later in Domain A.3 and A.4.
5. Continuous Systems (Distributed Mass)
When mass is spread over a volume (e.g., a rocket, satellite bus, or propellant tank), we model it as a continuum rather than discrete particles. Let the body occupy a region \(B\) with density \(\rho(\mathbf{r},t)\). A differential mass element is
\[ dm = \rho(\mathbf{r},t)\,dV. \]
Each material point has position \(\mathbf{r}\), velocity \(\mathbf{v}(\mathbf{r},t)\), and acceleration \(\mathbf{a}(\mathbf{r},t)\).
5.1 Kinematics and Velocity Field
The velocity field
\[ \mathbf{v} = \mathbf{v}(\mathbf{r},t) \]
associates a velocity with every point in the body. This viewpoint is essential for fluids, flexible structures (solar arrays, tethers), and fuel slosh inside tanks.
5.2 Kinetic Energy
The total kinetic energy of the continuous body is
\[ T = \frac{1}{2}\int_B \mathbf{v}\cdot\mathbf{v}\,dm. \]
For a rigid body, this integral separates into translational and rotational parts, leading to
\[ T = \frac{1}{2} M V_c^2 + \frac{1}{2}\boldsymbol{\omega}^T \mathbf{I}_c \boldsymbol{\omega}. \]
5.3 Linear Momentum
The total linear momentum is
\[ \mathbf{p} = \int_B \mathbf{v}\,dm. \]
For a rigid body, this reduces to \(\mathbf{p} = M\mathbf{V}_c\). If external forces vanish,
\[ \frac{d\mathbf{p}}{dt} = \mathbf{F}_{\text{ext}}, \]
so total momentum is conserved when \(\mathbf{F}_{\text{ext}} = \mathbf{0}\).
5.4 Angular Momentum
The angular momentum about a point \(P\) is
\[ \mathbf{H}_P = \int_B \mathbf{r}_{/P} \times \mathbf{v}\,dm, \]
with rate of change
\[ \frac{d\mathbf{H}_P}{dt} = \mathbf{L}_P, \]
where \(\mathbf{L}_P\) is the net external moment about \(P\). For rigid bodies, this leads directly to the inertia tensor and Euler’s rotational equations.
5.5 Control Volume Choice and Time Differentiation
If the control volume \(B\) is chosen to be the entire rigid body, the shape does not change and the boundary moves with the material. In body coordinates, \(B\) is effectively time-invariant, so for any field \(f\),
\[ \frac{d}{dt}\int_B f\,dm = \int_B \frac{df}{dt}\,dm. \]
If the body deforms, the domain of integration moves with time and we must use the Leibniz integral rule or the more general Reynolds Transport Theorem (RTT):
\[ \frac{d}{dt}\int_B f\,dm = \int_B \frac{\partial f}{\partial t}\,dm + \int_{\partial B} f (\mathbf{v}_{\text{rel}}\cdot\mathbf{n})\,dA, \]
where \(\mathbf{v}_{\text{rel}}\) is the velocity relative to the control surface and \(\mathbf{n}\) is the outward normal. The surface term accounts for mass and momentum crossing the boundary. Choosing a time-invariant control volume (rigid-body assumption) avoids this complication and yields compact classical equations.
5.6 Why Distributed Mass Models Matter in Aerospace
Continuous-mass modeling is essential for:
- propellant slosh in launch vehicles,
- flexible solar panels and booms,
- aeroelastic wings and lightweight structures,
- tethers and large deployable systems,
- atmospheric loads on re-entry vehicles,
- plume and mass-ejection phenomena.
It provides the bridge between particle mechanics, rigid-body dynamics, and advanced flight-dynamics models.
6. Rocket Equation and Specific Impulse
6.1 Tsiolkovsky Rocket Equation (No External Forces)
For a rocket that ejects mass with constant exhaust speed \(v_e\) (relative to the vehicle) and is subject to negligible external forces, the change in vehicle speed between initial mass \(m_0\) and final mass \(m_f\) is
\[ \Delta v = v_e \ln\left(\frac{m_0}{m_f}\right). \]
This highlights the importance of the mass ratio \(m_0/m_f\) and the exhaust velocity \(v_e\).
6.2 Thrust and Specific Impulse
If \(\dot{m}\) is the propellant mass flow rate and the exhaust pressure term is small, the thrust can be approximated by
\[ F_s \approx \dot{m}\,v_e. \]
The specific impulse \(I_{sp}\) is defined as
\[ I_{sp} = \frac{F_s}{\dot{m} g_0}, \]
where \(g_0\) is standard gravity at Earth’s surface. Higher \(I_{sp}\) means more thrust per unit propellant flow, effectively corresponding to a higher exhaust velocity. \(I_{sp}\) is a key performance measure when comparing rocket engines or propulsion technologies.