Impulse and Momentum in Collisions and Variable Mass Systems

1. The Impulse-Momentum Theorem

Let's begin, as always, with Newton's second law, $\vec{F}=m\vec{a}$.
We can rewrite acceleration as the time derivative of velocity, $\vec{a}=d\vec{v}/dt$. For a constant mass $m$, we can bring it inside the derivative:

Physicists define the quantity $m\vec{v}$ as the linear momentum, denoted by $\vec{p}$.

So, Newton's second law can be stated in what is actually its more general form: The net force on an object is equal to the rate of change of its linear momentum.

Now, let's integrate this equation with respect to time from an initial time $t_i$ to a final time $t_f$:

The integral on the right, by the Fundamental Theorem of Calculus, is simply the total change in momentum, ${\vec{p}}_f-{\vec{p}}_i=\Delta \vec{p}$.

The integral on the left is called the impulse, denoted by $\vec{J}$. It represents the total effect of the force acting over the time interval.

This brings us to the Impulse-Momentum Theorem:

The net impulse delivered to an object is equal to its change in momentum.

2. Conservation of Momentum

The Impulse-Momentum Theorem leads directly to one of the most important conservation laws in physics. If the net external force on a system of particles is zero (${\vec{F}}_{net,ext}=0$), then:

This implies that the total momentum of the system, ${\vec{p}}_{sys}$, is constant. This is the Principle of Conservation of Linear Momentum.

This principle is particularly powerful for analyzing collisions. During the brief moment of a collision, the internal forces between the colliding objects are often enormous compared to external forces like gravity or friction. These large, short-lived forces are called impulsive forces. The external forces are called non-impulsive. By ignoring the non-impulsive external forces, we can treat the total momentum of the colliding objects as being conserved just before and just after the collision.

The following video provides an excellent summary of these concepts, including the crucial distinction between impulsive and non-impulsive forces.

3. Analyzing Collisions

With the principle of momentum conservation in hand, we can analyze different types of collisions.

3.1. Classifying Collisions: The Coefficient of Restitution

While momentum is conserved in collisions (assuming negligible external forces), kinetic energy is not always conserved. The degree to which kinetic energy is conserved is characterized by the coefficient of restitution, $e$.

As you read, here is a summary of the collision types:

• Elastic Collision ($e=1$): Kinetic energy is conserved. $\Delta K=0$.
• Inelastic Collision ($0\le e<1$): Kinetic energy is lost, usually as heat, sound, or permanent deformation. $\Delta K<0$.
• Perfectly (or Totally) Inelastic Collision ($e=0$): The objects stick together after the collision, resulting in the maximum possible loss of kinetic energy (while still conserving momentum).
• Superelastic Collision ($e>1$): Kinetic energy increases. This happens if there's a release of stored potential energy, like in an explosion.

The coefficient of restitution $e$ is defined for a one-dimensional collision as:

This is a practical, kinematic quantity that describes the "bounciness" of the collision.

The following video segment provides a very clear, practical explanation of how to apply the formula for $e$ in different scenarios.

3.2. One-Dimensional Collisions

For any 1D collision between two masses $m_1$ and $m_2$, we can solve for the two unknown final velocities, $v_{1f}$ and $v_{2f}$, by setting up a system of two equations:

1. Conservation of Momentum: $m_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}$
2. Coefficient of Restitution: $v_{2f}-v_{1f}=e(v_{1i}-v_{2i})$

A famous special case is an elastic ($e=1$), head-on collision between two equal masses. In this case, the particles simply exchange their velocities.

3.3. Two-Dimensional (Oblique) Collisions

For collisions that are not head-on, momentum is conserved as a vector quantity. This means we must apply the conservation principle independently to each component (e.g., x and y).

For a 2D collision, the equations become:

1. x-momentum: $m_1{v}_{1ix}+m_2{v}_{2ix}=m_1{v}_{1fx}+m_2{v}_{2fx}$
2. y-momentum: $m_1{v}_{1iy}+m_2{v}_{2iy}=m_1{v}_{1fy}+m_2{v}_{2fy}$
3. Kinetic Energy (if elastic): $\frac{1}{2}{m}_1{v}_{1i}^2+\frac{1}{2}{m}_2{v}_{2i}^2=\frac{1}{2}{m}_1{v}_{1f}^2+\frac{1}{2}{m}_2{v}_{2f}^2$

An interesting result for a 2D elastic collision between two equal masses, where one is initially at rest, is that the angle between their final velocity vectors is always 90°. For a rigorous proof of this, which you might find interesting, you can refer to section 18.5.4 in the MIT "Collision Theory" notes (LINK) you looked at earlier.

4. Variable Mass Systems: Rocket Propulsion

We now turn to a classic engineering application where the mass of the system itself changes: a rocket. A rocket accelerates forward by expelling exhaust gases backward at high velocity. We can't apply $\vec{F}=m\vec{a}$ directly because $m$ is not constant. Instead, we must return to the more fundamental form, ${\vec{F}}_{net}=d\vec{p}/dt$.

Let's derive the equation of motion for a rocket. Consider a rocket of mass $m$ moving at velocity $v$ at time $t$. In a time interval $dt$, it ejects a small mass of fuel $d{m}_f$ with an exhaust velocity ${\vec{v}}_{ex}$ relative to the rocket. The rocket's mass decreases by $dm=-d{m}_f$, and its velocity increases to $\vec{v}+d\vec{v}$.

The following video provides a clear, step-by-step derivation of the resulting thrust force using the conservation of momentum.

The derivation shows that the expelling of gas creates a propulsive force, known as thrust ($\vec{T}$), on the rocket. If the mass is ejected at a rate $\dot{m}=dm/dt$ with a relative velocity $\vec{c}$ (often written as ${\vec{v}}_{ex}$ or $\vec{u}$), the thrust is:

Note that since the rocket loses mass, $dm/dt$ is negative, making the thrust vector $\vec{T}$ point in the same direction as the rocket's velocity (opposite to the exhaust velocity $\vec{c}$).

The full equation of motion for the rocket, including an external force like gravity (${\vec{F}}_{ext}$), is:

This is the fundamental rocket equation.

For a more formal and comprehensive treatment, including how to integrate this equation and an interesting engineering application, please study the following notes.

The key result from integrating the rocket equation in the absence of external forces is the Tsiolkovsky Rocket Equation:

This equation shows that the total change in velocity ($\Delta v$) depends only on the exhaust velocity ($c$) and the ratio of initial to final mass.

Conclusion

In this lesson, we have uncovered another deep principle derived from Newton's laws. By integrating $F=ma$ with respect to time, we developed the impulse-momentum framework.

Key Takeaways:

• The Impulse-Momentum Theorem (${\vec{J}}_{net}=\Delta \vec{p}$) relates the net impulse from a force to the change in an object's momentum.
• If the net external force on a system is zero, its total momentum is conserved. This is the key to analyzing collisions.
• Collisions are classified by the coefficient of restitution ($e$), which quantifies the conservation of kinetic energy.
• Variable mass systems, like rockets, are analyzed by applying momentum conservation to a system comprising the main body and the ejected mass. This leads to the rocket equation, where acceleration is driven by thrust.

Preview of the Next Lesson:
We have now explored integrating Newton's laws with respect to both time and position. We will now return to applying $\vec{F}=m\vec{a}$ directly but to more complex scenarios. In the next lesson, you will learn to solve constraint motion problems involving pulleys, strings, and inclined planes using Newtonian dynamics, which requires careful application of free-body diagrams and geometric constraints.