Linear Momentum, Collisions, and Impulse: Homework Study Notes

Linear Momentum and Collisions

Definition of Linear Momentum

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Linear Momentum (p): Defined as the product of an object's mass and its velocity.

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

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

• Conservation of Momentum: In a closed system with no external forces, the total momentum before an event equals the total momentum after the event.

Impulse and Momentum Change

Impulse is the change in momentum of an object when a force is applied over a time interval.

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

$\vec{J} = \Delta \vec{p} = \vec{F}_{\text{avg}} \Delta t$

• Impulse-Momentum Theorem: The impulse on an object is equal to the change in its momentum.

Collisions and Recoil

Collisions involve the interaction of two or more bodies, resulting in the exchange of momentum. Recoil occurs when an object moves in the opposite direction after an interaction, such as a child throwing a stone on ice.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not.

• Recoil Velocity: The velocity gained by an object as a result of an action, such as throwing another object.

Example: A child of mass $m_1$ throws a stone of mass $m_2$ with velocity $v_2$. The recoil velocity $v_1$ of the child (assuming frictionless surface):

$m_1 v_1 = -m_2 v_2 \implies v_1 = -\frac{m_2}{m_1} v_2$

Applications and Example Problems

Momentum at Maximum Height

When an object is thrown vertically upward, its momentum at maximum height is zero because its velocity is zero at that instant. At other points, momentum can be calculated using $p = m v$.

• Example: A 0.100-kg ball thrown upward with initial speed 15.0 m/s.

• At maximum height: $v = 0$, so $p = 0$.

• Halfway up: Use kinematic equations to find $v$ at halfway point, then $p = m v$.

Recoil on Frictionless Surfaces

When a person throws an object on a frictionless surface, both the person and the object move in opposite directions to conserve momentum.

• Example: A 40.0-kg child throws a 0.500-kg stone at 5.00 m/s. The child's recoil velocity is:

$v_{\text{child}} = -\frac{m_{\text{stone}}}{m_{\text{child}}} v_{\text{stone}}$

Comparing Momentum: Baseball vs. Bullet

Momentum can be compared between different objects by equating their momenta and solving for unknowns.

• Example: A pitcher claims to throw a baseball with as much momentum as a 30.0-g bullet moving at 500 m/s. For a baseball of mass 0.145 kg:

$p_{\text{bullet}} = m_{\text{bullet}} v_{\text{bullet}}$

$p_{\text{baseball}} = m_{\text{baseball}} v_{\text{baseball}}$

$\text{Set } p_{\text{baseball}} = p_{\text{bullet}} \implies v_{\text{baseball}} = \frac{m_{\text{bullet}} v_{\text{bullet}}}{m_{\text{baseball}}}$

Impulse and Force-Time Graphs

Impulse can be determined from the area under a force vs. time graph. If the force varies, calculate the area piecewise.

• Example: A softball is hit, and the force varies over time. The impulse equals the change in momentum.

Special Applications

Setting the Earth in Motion

When a person jumps, they exert a force on the Earth, causing it to move in the opposite direction. The effect is extremely small due to Earth's large mass.

• Model: Treat the Earth as a solid object. Use conservation of momentum.

• Calculation: $m_{\text{person}} v_{\text{person}} = M_{\text{Earth}} v_{\text{Earth}}$

• Result: $v_{\text{Earth}} = \frac{m_{\text{person}}}{M_{\text{Earth}}} v_{\text{person}}$ (very small value)

Elastic Potential Energy in Springs

Energy Stored in a Spring

When a spring is compressed or stretched, it stores elastic potential energy.

• Formula:

$U = \frac{1}{2} k x^2$

• k: Spring constant (N/m)

• x: Displacement from equilibrium (m)

Collisions with Springs

When blocks are pushed together with a spring and released, conservation of momentum and energy principles can be used to analyze their motion.

• Example: Two blocks of mass $M$ and $3M$ are pushed together with a spring and released. Use conservation of momentum to find final velocities.

Summary Table: Key Quantities in Momentum and Collisions

Quantity	Symbol	Formula	SI Unit
Linear Momentum	p	$p = m v$	kg·m/s
Impulse	J	$J = F \Delta t$	N·s (same as kg·m/s)
Elastic Potential Energy	U	$U = \frac{1}{2} k x^2$	J (Joules)

Additional info:

• These problems are typical of a college-level physics course covering linear momentum, impulse, collisions, and energy in mechanical systems.

• Students should be familiar with kinematic equations, conservation laws, and basic algebraic manipulation to solve these problems.