Momentum, Impulse, and Collisions

Learning Outcomes

This chapter covers the fundamental concepts of momentum, impulse, and collisions in physics. By the end of this chapter, students should be able to:

• Define the momentum of a particle and explain how net force causes its momentum to change.

• Identify when the total momentum of a system is conserved.

• Distinguish between elastic, inelastic, and completely inelastic collisions.

• Describe the center of mass of a system and analyze its motion.

• Analyze systems where the mass of an object changes as it moves, such as in rocket propulsion.

Introduction

In many real-world situations, such as hailstones shattering a roof, the forces involved are complex and not easily analyzed using Newton's second law alone. In such cases, the concepts of momentum and impulse, along with the conservation of momentum, provide powerful tools for solving problems involving collisions and other interactions.

Momentum and Newton's Second Law

Definition of Momentum

Momentum (p) is a vector quantity defined as the product of an object's mass and its velocity:

• Momentum has both magnitude and direction (same as velocity).

• SI unit: kilogram meter per second (kg·m/s).

Newton's Second Law in Terms of Momentum

Newton's second law can be expressed in terms of momentum as follows:

• The net external force on a particle equals the rate of change of its momentum.

• This form is especially useful when mass is not constant.

Impulse

Definition of Impulse

Impulse (J) is the product of the average force applied to an object and the time interval over which it acts:

• Impulse is a vector quantity, with the same direction as the force.

• SI unit: Newton-second (N·s), which is equivalent to kg·m/s.

Impulse from Variable Forces

When the force varies with time, impulse is given by the area under the force vs. time curve:

• The area under the curve represents the total impulse delivered.

• Short, large forces and long, small forces can deliver the same impulse if the area under the curve is equal.

Impulse-Momentum Theorem

The impulse-momentum theorem relates the impulse delivered to an object to its change in momentum:

• The change in momentum of a particle equals the impulse of the net force acting on it.

• Applications: Landing with bent knees increases stopping time, reducing the force on the body.

Momentum vs. Kinetic Energy

While both momentum and kinetic energy are related to motion, they describe different physical quantities:

• Kinetic Energy is a scalar quantity given by .

• Momentum is a vector quantity given by .

• Kinetic energy is associated with the work done to accelerate an object, while momentum is associated with the effect of force over time (impulse).

Conservation of Momentum

Isolated Systems

An isolated system is one in which the net external force is zero. In such systems, the total momentum is conserved:

• Example: Two astronauts pushing off each other in space.

Vector Nature of Momentum

• Momentum must be added using vector addition.

• Direction is crucial when applying conservation of momentum.

Types of Collisions

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved.

• Example: Collisions between billiard balls.

• During the collision, kinetic energy may be temporarily stored as potential energy (e.g., in compressed springs), but is fully recovered after the collision.

Inelastic and Completely Inelastic Collisions

In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is transformed into other forms (e.g., heat, deformation).

• In a completely inelastic collision, the colliding objects stick together after the collision.

• Example: Cars crumpling in a crash (energy absorbed in deformation).

Summary Table: Types of Collisions

Elastic Collisions in One Dimension

For two objects A and B (with masses and ), where B is initially at rest, the final velocities after an elastic collision are:

• If , A reverses direction, B barely moves.

• If , A stops, B moves with A's original speed.

• If , A slows slightly, B moves at nearly twice A's original speed.

Center of Mass

Definition and Calculation

The center of mass of a system is the point where the system's mass can be considered to be concentrated for the purpose of analyzing motion.

For particles with masses at positions :

• For symmetric objects (cube, sphere, cylinder), the center of mass is at the geometric center.

• For objects with an axis of symmetry (disk, donut), the center of mass lies along the axis, but may not be within the material itself.

Motion of the Center of Mass

The total momentum of a system equals the total mass times the velocity of the center of mass:

• The center of mass moves as if all external forces act at that point.

External Forces and Center-of-Mass Motion

• The motion of the center of mass is determined by the net external force on the system.

• If the net external force is zero, the center of mass moves at constant velocity.

Rocket Propulsion

In systems where mass changes (such as rockets burning fuel), the analysis of momentum must account for the ejected mass.

• As a rocket expels fuel, its mass decreases and its velocity increases to conserve momentum.

• Example: Atlas V launch vehicle ejects over 1000 kg of fuel per second at speeds of nearly 4000 m/s.