Momentum and Impulse


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

1. Concept of Momentum

  • an alternate variable (to velocity) to describe translational motion that is particularly powerful when describing the motion of a multi-body system.

  • The momentum of a system is unaffected by internal forces - forces between two objects that are both inside the system.

  • Momentum is changed only by external forces - forces on any part of the system that arise from interactions with objects outside of the system.

  • For systems that experience no external forces, momentum is constant (or conserved).

Momentum

  • Traditionally given the symbol $p$ (partly for "impulse", and since $m$ is already taken):

    $$\vec{p} = m\vec{\upsilon}$$ for a single particle of mass $m$ traveling with velocity $\upsilon$.

  • For a system composed of multiple particles, the momentum of the system is just the sum of the momenta of the masses in the system:

$$\vec{p}^{system} = \sum_i m_i\vec{\upsilon}_i$$

Impulse

Impulse (which is given the symbol $J$) is defined by:

$$\vec{J} = \int_{t_i}^{t_f} \sum_i \vec{F}^{ext}\;dt$$

for the net force acting over a time interval $t_i$ to $t_f$.

2. Collision (Linear Motions)

  • Forces

    • Internal forces act between the objects in the system

    • External forces act between objects in the system and the environment

  • Conservation of momentum in isolated systems

    • An isolated system is a collection of objects that may interact interanlly, but where the net external force on all the objects is zero. For an isolated system, the total momentume is conserved.

2.1. Complete Elastic Collision

$$\text{before}$$ $$\text{after}$$

  • needs two equations

    1) momentum conservation (system: $m_1 + m_2$)

    $$m_1 \upsilon_1 = m_1 \upsilon_1 ' + m_2 \upsilon_2'$$
    2) if complete elastic (i.e., kinetic energy conserved)

    $$ \frac{1}{2}m_1 \upsilon_1^2 = \frac{1}{2}m_1 \upsilon_1'^2 + \frac{1}{2}m_2 \upsilon_2'^2$$

  • general solution $$ \begin{align*} \upsilon_1'^2 & = \left( \frac{m_1 - m_2}{m_1 + m_2} \right)\upsilon_1 \\ \upsilon_2'^2 & = \left( \frac{2 m_1}{m_1 + m_2}\right)\upsilon_1 \qquad \text{positive direction} \end{align*} $$

    1) $m_1 \gg m_2 \;(m_2 \Rightarrow 0)$ $$ \begin{align*} \upsilon_1' & = \upsilon_1\\ \upsilon_2' & = 2 \upsilon_1 \qquad \text{not intuitive} \end{align*} $$
    2) $m_1 \ll m_2 \;(m_1 \Rightarrow 0)$ $$ \begin{align*} \upsilon_1' & = -\upsilon_1\\ \upsilon_2' & = 0 \end{align*} $$
    3) $m_1 = m_2$ $$ \begin{align*} \upsilon_1' & = 0 \\ \upsilon_2' & = \upsilon_1 \end{align*} $$
    4) $2 m_1 = m_2$ $$ \begin{align*} \upsilon_1' & = -\frac{1}{3}\upsilon_1 \\ \upsilon_2' & = \frac{2}{3}\upsilon_1 \end{align*} $$
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Collision simulator (optional)

  • processing
In [6]:
%%processing
int m1 = 20; 
int m2 = 10;
float v1 = 3;
float v2 = -3;

PVector x1;
PVector x2;

void setup() {
  size(700, 300);
  background(255);

  x1 = new PVector(200, height/2);
  x2 = new PVector(width-200, height/2);

  fill(100);
  ellipse(x1.x, x1.y, 2*m1, 2*m1);
  ellipse(x2.x, x2.y, 2*m2, 2*m2);
}

void draw() {
  background(255);

  x1.x += v1;
  x2.x += v2;

  fill(100);
  ellipse(x1.x, x1.y, 2*m1, 2*m1);
  ellipse(x2.x, x2.y, 2*m2, 2*m2);

  if  ((abs(x1.x-x2.x) <= m1+m2)) {

    float temp1 = ((m1-m2)*v1+2*m2*v2)/(m1 + m2);
    float temp2 = ((m2-m1)*v2+2*m1*v1)/(m2 + m1);
    v1 = temp1;
    v2 = temp2; 
  }
}

2.2. Question

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2.3. Complete Inelastic Collision

$$\text{before}$$ $$\text{after}$$

1) momentum conservation

$$m_1 \upsilon_1 = m_1 \upsilon_1 ' + m_2 \upsilon_2'$$

2) if complete inelastic

$$ \upsilon_1' = \upsilon_2' $$
  • general solution
$$ \begin{align*} \upsilon_1'^2 & = \left( \frac{m_1}{m_1 + m_2} \right)\upsilon_1 \\ \upsilon_2'^2 & = \left( \frac{m_1}{m_1 + m_2}\right)\upsilon_1 \end{align*} $$
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2.3. Explosion

  • initially a spring is compressed, then it is cut
    $$\text{before}$$ $$\text{after}$$

1) momentum conservation

$$0 + 0 = m_1 \upsilon_1 ' + m_2 \upsilon_2' \implies \frac{\upsilon_2'}{\upsilon_1'} = \frac{m_1}{m_2}$$

2) if mechanical energy conserved

$$\frac{1}{2}kx^2 = \frac{1}{2}m_1 \upsilon_1'^2 + \frac{1}{2}m_2 \upsilon_2'^2 $$
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Example 1: Momentum + Kinetic Energy


A pendulum consisting of a sphere of mass $m$ attached to the end of a massless rope of length $L$ is initially lifted by $h$ so that the rope is tight and horizontal, and released. At the bottom of its path, the sphere hits a block of mass $M = 2m$ lying at rest on a frictionless table. The collision is elastic.


Example 2: Momentum + Kinetic Energy

(a) Find the velocities of the sphere and the block immediately after the collision.

(b) How high does the pendulum swing after the collision?


Example 3: Super-ball

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In super-ball we take two balls, one small and one large, and release them together from a height $h_0$ above the ground, as illustrated in the below figure. What is the maximum height $h_1$ reached by the top ball after the collision? Assume that all collisions are conservative.


Example 4: Rocket

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$$\text{before}$$ $$\text{after}$$

Question

  • Think about how to escape from a frozen lake (no friction ice)

3. Impulse

Impulse (which is given the symbol $J$) is defined by:

$$\vec{J} = \int_{t_i}^{t_f} \sum_i \vec{F}^{ext}\;dt$$

for the net force acting over a time interval $t_i$ to $t_f$.

  • Impulse Causes Change in Momentum
$$\vec J = \Delta \vec p = \vec{m}_{t_f} - \vec{m}_{t_i} = \vec{F}_{\text{avg}} \,\delta t$$
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4. Angular Momentum

$$ \vec L = \vec r \times \vec p = I \vec{\omega}$$

  • Direction of angular momentum

  • It is also conserved when no external torque

  • Riding a bicycle

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