Momentum, Impulse, and Conservation Laws: Physics with Calculus Study Guide

Momentum and Impulse

Impulse: Definition and Physical Meaning

Impulse is a fundamental concept in physics describing the effect of a force acting over a short time interval, typically during collisions or rapid interactions. It quantifies the change in momentum resulting from the force.

• Impulse (J): The product of the average force and the time interval over which it acts. It is a vector quantity.

• Formula: $J = F_{\text{avg}} \Delta t$

• Units: Newton-seconds (N·s), equivalent to kg·m/s.

• Physical Example: Kicking a soccer ball, where the force rises quickly to a maximum and then falls back to zero as contact ends.

• Impulse is proportional to the area under the force vs. time curve.

Impulse and Force Curves

The effect of an impulsive force is determined by the area under the force-time curve. A harder kick (higher force) or longer contact time increases the impulse and the resulting speed of the object.

• Average Force (Favg): Defined as the constant force that would produce the same impulse over the same time interval.

• Formula: $F_{\text{avg}} = \frac{J}{\Delta t}$

Example: Impulse on a Bouncing Ball

For a triangular force curve, the impulse is calculated as the area of the triangle:

• Area of triangle: $\frac{1}{2} \times \text{height} \times \text{base}$

• Impulse: $J = \frac{1}{2} \times 300\,\text{N} \times 0.0080\,\text{s} = 1.2\,\text{N·s} = 1.2\,\text{kg·m/s}$

• Average force: $F_{\text{avg}} = \frac{1.2\,\text{N·s}}{0.0080\,\text{s}} = 150\,\text{N}$

Momentum and the Impulse-Momentum Theorem

Momentum: Definition and Properties

Momentum is the product of an object's mass and velocity, and is a vector quantity pointing in the direction of motion.

• Formula: $\vec{p} = m \vec{v}$

• Units: kg·m/s

• Momentum is a vector: $p_x = m v_x$, $p_y = m v_y$

Impulse-Momentum Theorem

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

• Formula: $\vec{J} = \Delta \vec{p} = m \vec{v}_f - m \vec{v}_i$

• Component form: $J_x = m (v_{fx} - v_{ix})$, $J_y = m (v_{fy} - v_{iy})$

Example: Change in Momentum (Ball Rebounding)

When a ball rebounds from a wall, its momentum changes direction. The impulse delivered by the wall equals this change.

• Initial momentum: $p_{ix} = 0.25\,\text{kg} \times 1.3\,\text{m/s} = 0.325\,\text{kg·m/s}$

• Final momentum: $p_{fx} = 0.25\,\text{kg} \times (-1.1\,\text{m/s}) = -0.275\,\text{kg·m/s}$

• Change in momentum: $\Delta p_x = p_{fx} - p_{ix} = -0.275 - 0.325 = -0.60\,\text{kg·m/s}$

• Impulse: $J_x = -0.60\,\text{kg·m/s}$ (negative indicates direction)

Conservation of Momentum

Law of Conservation of Momentum

In an isolated system (no net external force), the total momentum before an interaction equals the total momentum after.

• Formula: $\vec{P}_{\text{initial}} = \vec{P}_{\text{final}}$

• System: Defined by a boundary; internal forces do not change total momentum, only external forces do.

• Isolated system: Net external force is zero.

Example: Ice Skaters Pushing Off

Two skaters push off from each other on frictionless ice. The system is isolated, so momentum is conserved.

• Initial momentum: Both at rest, $P_i = 0$

• Final momentum: $m_S v_{Sf} + m_D v_{Df} = 0$

• Solving: $v_{Df} = -\frac{m_S}{m_D} v_{Sf}$

• Example values: $v_{Df} = -\frac{45}{80} \times 2.2 = -1.2\,\text{m/s}$

Collisions and Explosions

Inelastic Collisions

In a perfectly inelastic collision, objects stick together and move with a common final velocity. Momentum is conserved, but kinetic energy is not.

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

• Applications: Clay hitting the floor, bullet embedding in wood.

Explosions

In an explosion, particles move apart after a brief, intense interaction. If the system is isolated, total momentum is conserved.

• Formula: $m_1 v_{1f} + m_2 v_{2f} = 0$ (if initially at rest)

Momentum in Two Dimensions

Solving Two-Dimensional Collision Problems

When collisions occur in two dimensions, conservation of momentum must be applied to each component:

• Component equations:

  • $p_{1x,f} + p_{2x,f} = p_{1x,i} + p_{2x,i}$

  • $p_{1y,f} + p_{2y,f} = p_{1y,i} + p_{2y,i}$

• Example: Falcon and pigeon collision, where both x and y components are calculated and combined to find final speed and direction.

Angular Momentum

Definition and Conservation

Angular momentum is the rotational analog of linear momentum, defined for rotating objects.

• Formula: $L = I \omega$

• Conservation: If no net external torque acts, angular momentum is conserved: $L_i = L_f$

• Moment of inertia (I): Depends on mass distribution; changing I (e.g., by moving mass outward) changes angular velocity to conserve L.

Example: Merry-Go-Round Period Change

When a person moves outward on a rotating platform, the moment of inertia increases, causing angular velocity to decrease to conserve angular momentum.

• Formula: $I_i \omega_i = I_f \omega_f$

• Period: $T = \frac{2\pi}{\omega}$

Summary Table: Rotational vs. Linear Dynamics

Rotational Dynamics	Linear Dynamics
Torque $\tau_{\text{net}}$	Force $\vec{F}_{\text{net}}$
Moment of inertia $I$	Mass $m$
Angular velocity $\omega$	Velocity $\vec{v}$
Angular momentum $L = I \omega$	Linear momentum $\vec{p} = m \vec{v}$

Key Concepts and Problem-Solving Strategies

• Impulse-momentum theorem: $\vec{J} = \Delta \vec{p}$

• Conservation of momentum: Total momentum is constant in isolated systems.

• Conservation of angular momentum: Total angular momentum is constant if no net external torque acts.

• Before-and-after visual overview: Draw diagrams showing the system before and after interaction, define knowns and unknowns.