Impulse and Momentum

In mechanics, understanding how forces act over time is crucial for analyzing impacts, crashes, and explosions. This is the realm of impulse and momentum. While the Work-Energy theorem deals with forces acting over a distance, the Impulse-Momentum theorem deals with forces acting over time. This distinction is critical because in many real-world scenarios—like a car crash, a hammer striking a nail, or a jet engine providing thrust—the force is not constant and acts over a very short duration. We cannot easily measure the force or the distance during the impact, but we can measure the velocities before and after.

Center of Mass Frame

Center of Mass Frame Concepts

Before analyzing collisions, it is helpful to define the center of mass (CM) velocity. For a system of particles, the CM moves with a constant velocity if no external forces act on the system.
v⃗cm=Σmiv⃗iΣmi=P⃗totalMtotal \vec{v}_{cm} = \frac{\Sigma m_i \vec{v}_i}{\Sigma m_i} = \frac{\vec{P}_{total}}{M_{total}}

Center of Mass Frame Concepts

In the CM reference frame, the total momentum of the system is always identically zero. This makes analyzing complex collisions (especially in 2D or 3D) mathematically much simpler.

Linear Momentum (pp)

Linear Momentum (pp) Concepts

Momentum is a measure of "how hard it is to stop" an object. It depends on both how massive the object is and how fast it's moving.

Linear Momentum (p⃗\vec{p})

The product of an object's mass and its velocity. It is a vector quantity, pointing in the same direction as the velocity. The SI unit is kg⋅m/s\text{kg} \cdot \text{m/s}.
p⃗=mv⃗ \vec{p} = m\vec{v}

Linear Momentum (pp) Concepts

Newton actually formulated his Second Law in terms of momentum, not acceleration. The net force on an object is equal to the rate of change of its momentum.
ΣF⃗=dp⃗dt=d(mv⃗)dt \Sigma \vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}

Linear Momentum (pp) Concepts

If mass is constant, this simplifies to our familiar F=maF=ma. However, this general form is powerful because it allows us to analyze systems where mass is changing, like a rocket burning fuel.

Impulse (JJ)

Impulse (JJ) Concepts

If momentum is the "amount of motion," then impulse is the "change in motion" caused by a force acting over a time interval.

Impulse (J⃗\vec{J})

The integral of a force over the time interval during which it acts. It is a vector quantity, pointing in the same direction as the force. The SI unit is N⋅s\text{N} \cdot \text{s} (which is equivalent to kg⋅m/s\text{kg} \cdot \text{m/s}).
J⃗=∫titfF⃗(t) dt \vec{J} = \int_{t_i}^{t_f} \vec{F}(t) \, dt

Impulse (JJ) Concepts

For a constant force, this simplifies to:
J⃗=F⃗avgΔt \vec{J} = \vec{F}_{avg} \Delta t

Impulse (JJ) Concepts

Graphically, impulse is the area under a Force vs. Time curve. This is why airbags and crumple zones save lives in car crashes. They increase the time (Δt\Delta t) over which a passenger's momentum is reduced to zero. By increasing the time, the average force (F⃗avg\vec{F}_{avg}) required to deliver the necessary impulse (J⃗\vec{J}) decreases significantly.

The Impulse-Momentum Theorem

The Impulse-Momentum Theorem Concepts

Just as the Work-Energy theorem connects work and kinetic energy, the Impulse-Momentum theorem connects impulse and momentum.

The Impulse-Momentum Theorem

The net impulse exerted on a particle is equal to the change in the particle's momentum.
J⃗net=Δp⃗=p⃗f−p⃗i=mv⃗f−mv⃗i \vec{J}_{net} = \Delta \vec{p} = \vec{p}_f - \vec{p}_i = m\vec{v}_f - m\vec{v}_i

The Impulse-Momentum Theorem Concepts

This theorem is essentially Newton's Second Law rewritten to focus on the time interval over which a force acts.

Conservation of Linear Momentum

Conservation of Linear Momentum Concepts

The true power of momentum lies in analyzing systems of interacting particles. When two objects collide, the forces they exert on each other are internal to the system. By Newton's Third Law, these forces are equal and opposite, so the impulses they impart to each other are also equal and opposite (J⃗1 on 2=−J⃗2 on 1\vec{J}_{1 \text{ on } 2} = -\vec{J}_{2 \text{ on } 1}).
This leads to a profound conclusion.

Conservation of Linear Momentum

If the net external force on a system is zero (ΣF⃗ext=0\Sigma \vec{F}_{ext} = 0), then the total linear momentum of the system is conserved (remains constant).
P⃗total,i=P⃗total,f \vec{P}_{total, i} = \vec{P}_{total, f} m1v⃗1i+m2v⃗2i=m1v⃗1f+m2v⃗2f m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}

Important

Momentum is a vector quantity. This means momentum must be conserved in each spatial dimension independently. In a 2D collision on an x-y plane:
Σpxi=ΣpxfΣpyi=Σpyf \begin{aligned} \Sigma p_{xi} &= \Sigma p_{xf} \\ \Sigma p_{yi} &= \Sigma p_{yf} \end{aligned}

Variable Mass Systems: Rocket Equation

Variable Mass Systems: Rocket Equation Concepts

A classic application of the generalized Newton's Second Law (ΣF⃗=dp⃗dt\Sigma \vec{F} = \frac{d\vec{p}}{dt}) is analyzing systems where mass is continuously ejected or accumulated, such as a rocket.
A rocket accelerates by ejecting mass (exhaust gas) backwards at a high relative velocity (vev_e). Conservation of momentum gives the Tsiolkovsky rocket equation, which relates the change in velocity (Δv\Delta v) to the effective exhaust velocity (vev_e) and the initial (mim_i) and final (mfm_f) mass of the rocket:
Δv=veln⁡(mimf) \Delta v = v_e \ln\left(\frac{m_i}{m_f}\right)

Variable Mass Systems: Rocket Equation Concepts

The term dmdtve\frac{dm}{dt} v_e is known as thrust, the upward force exerted on the rocket by the expelled exhaust.

Types of Collisions

Types of Collisions Concepts

While total momentum is always conserved in a collision (if external forces are negligible), kinetic energy is not necessarily conserved. Collisions are classified by what happens to the total kinetic energy.
  • Elastic Collisions: Both momentum and total kinetic energy are conserved. The objects bounce off each other perfectly without any loss of mechanical energy to heat or deformation. Example: Billiard balls (approximately), gas molecules.
  • Inelastic Collisions: Momentum is conserved, but kinetic energy is not. Some kinetic energy is converted into internal energy (heat, sound, deformation). Most real-world macroscopic collisions are inelastic.
  • Perfectly Inelastic Collisions: A specific type of inelastic collision where the objects stick together and move with a common final velocity after the impact. This results in the maximum possible loss of kinetic energy (though momentum is still conserved).

Coefficient of Restitution (ee)

Coefficient of Restitution (ee) Concepts

The degree of elasticity in a 1D collision is quantified by the coefficient of restitution, ee. It relates the relative speeds of approach and separation.

Coefficient of Restitution (ee)

The ratio of the relative speed of separation after collision to the relative speed of approach before collision.
e=v2f−v1fv1i−v2i e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}

Coefficient of Restitution (ee) Concepts

  • e=1e = 1: Perfectly elastic collision.
    • 0<e<10 < e < 1: Inelastic collision.
    • e=0e = 0: Perfectly inelastic collision (objects stick together, v1f=v2fv_{1f} = v_{2f}).

Collisions in Two Dimensions

Vector Conservation

In two-dimensional collisions, such as a glancing blow in billiards or a vehicle intersection crash, momentum is conserved independently in both the xx and yy directions. The principles are the same as 1D collisions, but require the use of vector components.

2D Momentum Conservation Equations

The separated x and y component equations for conservation of linear momentum.

$$ \begin{aligned} \sum p_{xi} &= \sum p_{xf} \implies m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx} \\ \sum p_{yi} &= \sum p_{yf} \implies m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy} \end{aligned} $$
Key Takeaways
  • Momentum (p⃗=mv⃗\vec{p} = m\vec{v}) is a vector describing the "amount of motion." The net force equals the rate of change of momentum.
  • Impulse (J⃗=∫F⃗dt\vec{J} = \int \vec{F} dt) is a vector describing the "change in motion." It equals the change in momentum (J⃗=Δp⃗\vec{J} = \Delta\vec{p}).
  • The Conservation of Momentum principle states that the total momentum of a system is constant if no net external forces act on it.
  • In Elastic collisions, both momentum and kinetic energy are conserved. In Inelastic collisions, only momentum is conserved; kinetic energy is lost. The Coefficient of Restitution quantifies this.
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