Course Notes š
Linear Momentum (\( p \))
The linear momentum of an object is the product of its mass and velocity. It is a vector quantity.
\[ p = mv \]
SI unit: kilogram meter per second (kgā m/s)
Newton's Second Law and Momentum
The net force acting on an object is equal to the rate of change of its momentum.
\[ F_{net} = \frac{\Delta p}{\Delta t} \]
Impulse (\( I \))
The impulse delivered to an object is the product of the average force acting on it and the time interval over which the force acts.
\[ I = F_{av} \Delta t \]
Impulse-Momentum Theorem
The impulse delivered to an object is equal to the change in its momentum.
\[ I = \Delta p = mv_f - mv_i \]
Example: Car Collision
A car of mass \( 1.50 \times 10^3 \) kg hits a wall. Its initial velocity is \( v_i = -15.0 \) m/s and its final velocity is \( v_f = 2.60 \) m/s. The collision lasts for \( 0.150 \) s.
(a) Impulse delivered to the car:
\( \Delta p = m(v_f - v_i) = (1.50 \times 10^3)(2.60 - (-15.0)) = 2.64 \times 10^4 \text{ kg m/s} \)(b) Average force on the car:
\( F_{av} = \frac{\Delta p}{\Delta t} = \frac{2.64 \times 10^4}{0.150} = 1.76 \times 10^5 \text{ N} \)
When no net external force acts on a system, the total momentum of the system remains constant in time. This means the initial momentum of the system is equal to its final momentum.
\[ p_i = p_f \] \[ m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} \]
Example: Archer on Ice
An archer (total mass 60.0 kg) is at rest on frictionless ice. He fires a 0.030 kg arrow horizontally at 50.0 m/s. Find his recoil velocity.
Let the archer be object 1 and the arrow be object 2. Initially, the total momentum is 0.
Conservation of momentum: \( p_i = p_f \)
\( 0 = m_1v_{1f} + m_2v_{2f} \)
\( v_{1f} = -\frac{m_2v_{2f}}{m_1} = -\frac{(0.030)(50.0)}{59.97} = -0.0250 \text{ m/s} \)
Types of Collisions
Inelastic Collisions: Momentum is conserved, but kinetic energy is not. Most real-world collisions are in this category.
Perfectly Inelastic Collisions: A type of inelastic collision where the objects stick together after colliding. Kinetic energy loss is at a maximum.
\[ m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f \]
Elastic Collisions: Both momentum and kinetic energy are conserved. This often happens at the atomic level or with objects like billiard balls.
\[ v_{1i} - v_{2i} = -(v_{1f} - v_{2f}) \]