### 8.1 Linear Momentum and Force

- Linear momentum (
for brevity) is defined as the product of a system’s mass multiplied by its velocity.*momentum* - In symbols, linear momentum $\mathbf{p}$ is defined to be
$$\mathbf{p}=m\mathbf{v},$$where
*$m$*is the mass of the system and $\mathbf{v}$ is its velocity. - The SI unit for momentum is $\text{kg}\xb7\text{m/s}$.
- Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
- In symbols, Newton’s second law of motion is defined to be
$${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta}\mathbf{p}}{\mathrm{\Delta}t}\text{,}$$${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta}\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta}t$ is the change time.

### 8.2 Impulse

- Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:
$$\mathrm{\Delta}\mathbf{p}={\mathbf{F}}_{\text{net}}\mathrm{\Delta}t.$$
- Forces are usually not constant over a period of time.

### 8.3 Conservation of Momentum

- The conservation of momentum principle is written
$${\mathbf{p}}_{\text{tot}}=\text{constant}$$or$${\mathbf{\text{p}}}_{\text{tot}}={\mathbf{\text{p}}\prime}_{\text{tot}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}(\text{isolated system}),$$${\mathbf{p}}_{\text{tot}}$ is the initial total momentum and ${\mathbf{\text{p}}\prime}_{\text{tot}}$ is the total momentum some time later.
- An isolated system is defined to be one for which the net external force is zero $\left({\mathbf{\text{F}}}_{\text{net}}=0\right)\text{.}$
- During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
- Conservation of momentum applies only when the net external force is zero.
- The conservation of momentum principle is valid when considering systems of particles.

### 8.4 Elastic Collisions in One Dimension

- An elastic collision is one that conserves internal kinetic energy.
- Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.

### 8.5 Inelastic Collisions in One Dimension

- An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).
- A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.
- Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.

### 8.6 Collisions of Point Masses in Two Dimensions

- The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the $x$-axis parallel to the velocity of the incoming particle.
- Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the $x$-axis), stated by ${m}_{1}{v}_{1}={m}_{1}{v\prime}_{1}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta}_{1}+{m}_{2}{v\prime}_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta}_{2}$ and along the direction perpendicular to the initial direction (the $y$-axis) stated by $0={m}_{1}{v\prime}_{1y}+{m}_{2}{v\prime}_{2y}$.
- The internal kinetic before and after the collision of two objects that have equal masses is $$\frac{1}{2}{{\text{mv}}_{1}}^{2}=\frac{1}{2}{{\text{mv}\prime}_{1}}^{2}+\frac{1}{2}{{\text{mv}\prime}_{2}}^{2}+{\text{mv}\prime}_{1}{v\prime}_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\left({\theta}_{1}-{\theta}_{2}\right).$$
- Point masses are structureless particles that cannot spin.

### 8.7 Introduction to Rocket Propulsion

- Newton’s third law of motion states that to every action, there is an equal and opposite reaction.
- Acceleration of a rocket is $a=\frac{{v}_{\text{e}}}{m}\phantom{\rule{0.25em}{0ex}}\frac{\text{\Delta}m}{\text{\Delta}t}-g$.
- A rocket’s acceleration depends on three main factors. They are
- The greater the exhaust velocity of the gases, the greater the acceleration.
- The faster the rocket burns its fuel, the greater its acceleration.
- The smaller the rocket's mass, the greater the acceleration.