Chapter 9: Momentum – Impulse, Collisions, and Conservation Laws

Momentum and Impulse

Impulse

Impulse is a vector quantity that measures the effect of a force acting over a time interval. It is defined as the product of the average force and the time interval during which the force acts:

• Definition: $\vec{I} \equiv \vec{F} \Delta t$

• SI Units: kg·m/s

• Impulse in Multiple Dimensions:

  • One dimension: $I_x = F_x \Delta t$

  • Two dimensions: $I_y = F_y \Delta t$

  • Three dimensions: $I_z = F_z \Delta t$

• Impulse as Area: The impulse delivered by a force is equal to the area under the force vs. time curve.

Average Force in Impulse

The average force, $F_{av}$, is the constant force that would produce the same impulse as the actual time-varying force over the same interval:

• $F_{av} \Delta t = \Delta p$

Momentum

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

• Definition: $\vec{p} \equiv m \vec{v}$

• SI Units: kg·m/s

• Relation to Kinetic Energy: $KE = \frac{p^2}{2m}$

Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse delivered to an object equals the change in its momentum:

• $\vec{F} \Delta t = \vec{I} = \Delta \vec{p} = m v_f - m v_i$

Solving Impulse and Momentum Problems

Example: Car Collision

Consider a car of mass $1.50 \times 10^3$ kg colliding with a wall and rebounding. The initial and final velocities are $v_i = -15.0$ m/s and $v_f = 2.60$ m/s, respectively. If the collision lasts for 0.150 s, the impulse and average force can be calculated as follows:

• Impulse: $I = m v_f - m v_i$

• Average Force: $F_{av} = \frac{\Delta p}{\Delta t}$

Force-Time Graphs in Collisions

In real collisions, the force varies with time. The area under the force-time graph gives the impulse delivered to the object.

Conservation of Momentum

Principle of Conservation

When no net external force acts on a system, the total momentum of the system remains constant:

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

Newton's Third Law and Collisions

During a collision, the forces that two objects exert on each other are equal in magnitude and opposite in direction, as per Newton's third law:

• $\vec{F}_{12} = -\vec{F}_{21}$

Elastic and Inelastic Collisions

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. Before the collision, the objects move independently; after the collision, their velocities change, but the total energy and momentum remain constant.

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

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

Demonstration: Newton's Cradle

Newton's cradle demonstrates conservation of momentum and energy in elastic collisions. When one ball strikes the group, one ball exits on the opposite side with nearly the same velocity.

Inelastic Collisions

In an inelastic collision, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, the objects stick together after the collision and move with a common velocity.

• Momentum Conservation: $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f$

• Kinetic Energy: Not conserved; some is transformed into other forms (e.g., heat, deformation).

Momentum and Collisions in Two Dimensions

Two-Dimensional Collisions

In two-dimensional collisions, the conservation of momentum applies separately to each component (x and y):

• x-component: $m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}$

• y-component: $m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy}$

Example: Collision at an Intersection

A car and a pickup truck collide at an intersection. The conservation of momentum in both x and y directions allows calculation of the velocity and direction of the combined wreckage after a perfectly inelastic collision.

Key Terms Table

Term	Definition
Impulse	Change in momentum due to a force acting over a time interval
Momentum	Product of mass and velocity of an object
Elastic Collision	Collision where both momentum and kinetic energy are conserved
Inelastic Collision	Collision where momentum is conserved but kinetic energy is not
Perfectly Inelastic Collision	Collision where objects stick together after impact
Angular Momentum	Rotational analog of linear momentum (not detailed in this chapter)

Additional info: Angular momentum is mentioned but not elaborated in these notes; it is typically covered in detail in a separate chapter.