Comparison and Dynamics of Momentum and Kinetic Energy

Momentum and kinetic energy are distinct physical quantities used to describe the motion of an object.
Both quantities are functions of an object's mass ( $m$ ) and its velocity ( $v$ ).
CLM 9: A fundamental similarity between these two quantities is that momentum and kinetic energy change in the same way when an object's velocity changes. Specifically, if the magnitude of the velocity increases, both quantities increase; if the velocity decreases, both quantities decrease; and if the velocity is zero, both quantities are zero.

Linear Momentum ( $p$ ): The product of an object's mass and its velocity. $p = m \times v$
- Units: The SI units for momentum are kilogram-meters per second ( $\text{kg}\,\text{m/s}$ ).
Kinetic Energy ( $K$ ): The energy an object possesses due to its motion. $K = \frac{1}{2} \times m \times v^2$
- Units: The SI unit for kinetic energy is the Joule ( $J$ ), where $1\,\text{J} = 1\,\text{kg}\,\text{m}^2/\text{s}^2$ .

SIQ 9 asks how momentum and kinetic energy differ in their dependence on velocity:
Linear Relationship (Momentum): Momentum is directly proportional to velocity ( $p \propto v$ ). When mass is constant, any change in velocity results in a proportional change in momentum. For instance, if velocity doubles ( $2 \times v$ ), the momentum also doubles.
Quadratic Relationship (Kinetic Energy): Kinetic energy is proportional to the square of the velocity ( $K \propto v^2$ ). This means that energy grows much faster than momentum as speed increases. For example, if velocity doubles ( $2 \times v$ ), the kinetic energy increases by a factor of four ( $2^2 = 4$ ). If velocity triples ( $3 \times v$ ), the kinetic energy increases by a factor of nine ( $3^2 = 9$ ).

Momentum is a Vector Quantity:
- Momentum has both magnitude and direction. The direction of an object's momentum is identical to the direction of its velocity vector.
- To find the total momentum of a system of multiple objects, one must use vector addition, taking into account the direction of each moving part.
Kinetic Energy is a Scalar Quantity:
- Kinetic energy represents a magnitude only and does not have a direction.
- The square of the velocity in the kinetic energy formula removes the directional component of the velocity vector. Therefore, an object moving north at $10\,\text{m/s}$ has the same kinetic energy as an identical object moving south at $10\,\text{m/s}$ .
- The total kinetic energy of a system is simply the algebraic sum of the individual kinetic energies of all objects in the system.

Momentum Conservation: Momentum is conserved in all types of collisions (elastic, inelastic, and perfectly inelastic) as long as no external forces act on the system.
Kinetic Energy Conservation: Kinetic energy is NOT conserved in all collisions. It is conserved ONLY in perfectly elastic collisions. In most real-world (inelastic) collisions, some kinetic energy is converted into other forms of energy, such as thermal energy, sound, or the internal energy required to deform the colliding objects.

Dependency on Speed: Momentum is linear regarding speed; kinetic energy is quadratic.
Directionality: Momentum accounts for direction (vector); kinetic energy does not (scalar).
Conservation: Momentum is always conserved in an isolated system during a collision; kinetic energy is only conserved in specific (elastic) collision scenarios.
Mathematical Link: Kinetic energy can be calculated from momentum and mass using the following formula: $K = \frac{p^2}{2 \times m}$

SIQ 9: How do momentum and kinetic energy differ in the way they depend on an object's velocity?
- Momentum depends on the first power of velocity ( $v^1$ ), while kinetic energy depends on the square of the velocity ( $v^2$ ). This results in a linear relationship for momentum and a quadratic relationship for kinetic energy.
How else do these two quantities differ?
- A primary difference is that momentum is a vector quantity (magnitude and direction), whereas kinetic energy is a scalar quantity (magnitude only). Additionally, their conservation properties in collisions differ; momentum is always conserved in closed systems, while kinetic energy is conserved only in perfectly elastic collisions. Finally, they use different units: $\text{kg}\,\text{m/s}$ for momentum and Joules ( $J$ ) for kinetic energy.