Linear Momentum and Collisions: Study Notes

Linear Momentum and Collisions

Introduction

This chapter covers the fundamental concepts of linear momentum, impulse, the conservation of momentum, and collisions. These principles are essential for understanding the motion and interactions of objects in classical physics.

Momentum

Definition of Momentum

Momentum is a measure of the motion of an object and is defined as the product of its mass and velocity. The symbol for momentum is p.

• Formula: $\vec{p} = m \vec{v}$

• SI Units: kg·m/s

• Linear Momentum: Sometimes referred to as linear momentum to distinguish it from angular momentum, which is associated with rotating objects.

Properties of Momentum

• Vector Quantity: Momentum has both magnitude and direction. Its direction is the same as the velocity vector.

• Change in Momentum: An object's momentum changes whenever its mass or velocity changes.

Example: Comparing Momentum

Consider catching a slow, heavy object versus a fast, lightweight object. If both have the same momentum, the effect of catching them can be similar.

• Beanbag Example: A 1-kg beanbag moving downward at 4 m/s comes to rest on the floor. $\Delta \vec{p} = \vec{p}_f - \vec{p}_i = 0 - (-4\,\text{kg} \cdot \text{m/s}) = 4\,\text{kg} \cdot \text{m/s}$

• Rubber Ball Example: A 1-kg rubber ball moving downward at 4 m/s bounces upward with the same speed. $\Delta \vec{p} = \vec{p}_f - \vec{p}_i = (4\,\text{kg} \cdot \text{m/s}) - (-4\,\text{kg} \cdot \text{m/s}) = 8\,\text{kg} \cdot \text{m/s}$

Impulse

Definition of Impulse

Impulse is the product of a force and the time interval over which it acts. It quantifies the effect of a force acting over a period of time.

• Formula: $\vec{I} = \vec{F} \Delta t$

• SI Units: N·s = kg·m/s

• Vector Quantity: Impulse points in the same direction as the force.

Impulse-Momentum Theorem

Impulse is directly related to the change in momentum of an object.

• Formula: $\vec{I} = \Delta \vec{p}$ $\vec{F} \Delta t = \vec{p}_f - \vec{p}_i$

• Application: A small force acting over a long time can produce the same impulse as a large force acting over a short time.

Example: Calculating Impulse

• Shopping Cart Example: A 22-kg cart is pushed with a force of 6.5 N for 1.9 s. $I = F \Delta t = (6.5\,\text{N})(1.9\,\text{s}) = 12.35\,\text{kg} \cdot \text{m/s}$

Conservation of Momentum

Principle of Conservation

If the total external force acting on a system is zero, the total momentum of the system remains constant.

• Formula: $\vec{p}_i = \vec{p}_f$

• Internal vs. External Forces: Only external forces can change a system's total momentum. Internal forces (action-reaction pairs) always sum to zero and do not affect the system's total momentum.

Applications

• Billiards: Momentum is transferred between balls during collisions, but the total momentum remains constant.

• Jumping: When you jump, the upward momentum you gain is balanced by the downward momentum of the Earth.

• Universe: Momentum conservation applies to the largest possible system—the universe. For example, an exploding star sends material in opposite directions, conserving total momentum.

Recoil

Recoil is the backward motion caused by two objects pushing off one another, such as a gun firing or a firefighter using a hose. Recoil is a direct result of momentum conservation.

Collisions

Types of Collisions

Collisions are classified based on whether kinetic energy is conserved.

• Elastic Collision: Kinetic energy is conserved. $KE_{f} = KE_{i}$

• Inelastic Collision: Kinetic energy is not conserved; some is converted to other forms such as heat or sound. $KE_{f} < KE_{i}$

• Completely Inelastic Collision: Colliding objects stick together after the collision.

Momentum Conservation in Collisions

Momentum is always conserved in collisions, regardless of the type.

• Formula for Two Objects: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Elastic Collision Equations

For elastic collisions, both momentum and kinetic energy are conserved, leading to two equations:

• Momentum Conservation: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Kinetic Energy Conservation: $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

Solving for Final Velocities

• Final Velocity of Object 1: $v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} + \frac{2m_2}{m_1 + m_2} v_{2i}$

• Final Velocity of Object 2: $v_{2f} = \frac{2m_1}{m_1 + m_2} v_{1i} + \frac{m_2 - m_1}{m_1 + m_2} v_{2i}$

Example: Completely Inelastic Collision

• When two railroad cars of equal mass collide and stick together, the final speed is half the initial speed, and half the initial kinetic energy is converted to other forms.

• Final Kinetic Energy: $KE_f = \frac{1}{2} m (\frac{v}{2})^2 = \frac{1}{4} m v^2 = \frac{1}{2} KE_i$

Real-World Collisions

• Most everyday collisions are inelastic, but objects like billiard balls approximate elastic collisions due to minimal deformation.

Summary Table: Types of Collisions

Type of Collision	Kinetic Energy Conserved?	Objects Stick Together?
Elastic	Yes	No
Inelastic	No	No
Completely Inelastic	No	Yes

Key Takeaways

• Momentum is always conserved in isolated systems.

• Impulse changes an object's momentum and is equal to the force times the time interval.

• Collisions can be elastic or inelastic, depending on whether kinetic energy is conserved.

• Understanding these principles is crucial for analyzing motion and interactions in physics.