3.9 Linear Momentum

Linear momentum is a fundamental concept in physics that describes an object's "quantity of motion." It is a measure of how difficult it is to stop or change the direction of a moving object. Often referred to simply as momentum, it is a vector quantity that combines an object's mass and its velocity into a single value.

Definition and Formula

Linear momentum ($\vec{p}$) is defined as the product of an object's mass ($m$) and its velocity ($\vec{v}$).

Formula: $\vec{p}=m\vec{v}$

• Vector Quantity: Momentum is a vector. Its direction is the same as the direction of the object's velocity.
• Dependence: The magnitude of momentum is directly proportional to both the mass and the velocity of the object. A heavy object or a fast-moving object will have a large momentum.

Scalar and Vector Quantities→

Units and Dimensions

• SI Unit: The standard unit for momentum is the kilogram-meter per second (kg·m/s).
• Equivalent Unit: It can also be expressed in Newton-seconds (N·s).
  • Proof: $\text{N}\cdot \text{s}=(\text{kg}\cdot \text{m}\cdot {\text{s}}^{-2})\cdot \text{s}=\text{kg}\cdot \text{m}\cdot {\text{s}}^{-1}$
• Dimensions: The dimensional formula for momentum is $[ML{T}^{-1}]$.

Dimensional Homogeneity Check for $\vec{p}=m\vec{v}$: $[\vec{p}]=[m][\vec{v}]=[M][L{T}^{-1}]=[ML{T}^{-1}]✓$ Both sides have the same dimensions, confirming the equation is dimensionally homogeneous.

Derived Units→ Dimensions→

Momentum and Newton's Second Law of Motion

Newton's Second Law of Motion is most accurately and generally stated in terms of momentum.

Statement:

Mathematical Form: ${\vec{F}}_{net}=\frac{\Delta \vec{p}}{\Delta t}$ Where:

• $\Delta \vec{p}$ is the change in momentum.
• $\Delta t$ is the time interval over which the change occurs.

Derivation to F = ma: If the mass ($m$) of the object is constant, the change in momentum is: $\Delta \vec{p}=\Delta (m\vec{v})=m(\Delta \vec{v})=m({\vec{v}}_f-{\vec{v}}_i)$ Substituting this into the force equation: ${\vec{F}}_{net}=\frac{m({\vec{v}}_f-{\vec{v}}_i)}{\Delta t}$ Since acceleration is defined as $\vec{a}=\frac{{\vec{v}}_f-{\vec{v}}_i}{\Delta t}$, this simplifies to: ${\vec{F}}_{net}=m\vec{a}$ This shows that $F=ma$ is a special case of the more fundamental momentum principle.

Impulse

In many cases, a large force acts for a very short interval of time, such as a bat hitting a ball. The product of force and time is called impulse.

Formula: $\vec{I}={\vec{F}}_{avg}\times \Delta t=\Delta \vec{p}=m{\vec{v}}_f-m{\vec{v}}_i$

The Impulse-Momentum Theorem states that the impulse acting on a body equals the change in its momentum. This is why:

• Catching a cricket ball with a moving hand (increasing $\Delta t$) reduces the average force.
• Crumple zones in cars increase collision time, reducing the force on passengers.

Law of Conservation of Linear Momentum

The total linear momentum of an isolated system (one on which no net external force acts) remains constant in both magnitude and direction.

Mathematical Statement (for two colliding bodies): $m_1{\vec{v}}_1+m_2{\vec{v}}_2=m_1{\vec{v}}_1^{\prime }+m_2{\vec{v}}_2^{\prime }$ where ${\vec{v}}_1,{\vec{v}}_2$ are initial velocities and ${\vec{v}}_1^{\prime },{\vec{v}}_2^{\prime }$ are final velocities.

Basis: This law follows from Newton's Third Law combined with Newton's Second Law. During a collision, the action-reaction forces are equal and opposite, producing equal and opposite changes in momentum, so the total remains constant.

Summary Table