Chapter 1. Momentum and impulse FREE
1.1 What is momentum?
Momentum is the property of a moving object that indicates how difficult it is to bring the object to rest. It depends on both the mass and the velocity of the object. We calculate linear momentum with \( p = mv \), where \( m \) is the mass in kilograms and \( v \) is the velocity in \( \text{m·s}^{-1} \). Momentum is a vector quantity: it has both magnitude and direction, and the direction is the same as the direction of the velocity. The SI unit of momentum is \( \text{kg·m·s}^{-1} \).
Example: A truck of \( 2000 \) kg moves at \( 15 \) \( \text{m·s}^{-1} \) eastward. Its momentum is \( p = mv = 2000 \times 15 = 30\,000 \) \( \text{kg·m·s}^{-1} \) eastward. A bicycle with a smaller mass at the same velocity has much less momentum, and it is therefore easier to stop the bicycle.
1.2 Impulse
When a net force \( F \) acts on an object for a time interval \( \Delta t \), the object's momentum changes. We call the product of force and time the impulse. From Newton's second law the impulse-momentum theorem follows:
\( F\Delta t = \Delta p = m v_f - m v_i \)
The unit of impulse is \( \text{N·s} \), which is exactly the same as \( \text{kg·m·s}^{-1} \). This relationship explains why safety equipment works. If we extend the contact time \( \Delta t \) while the change in momentum stays the same, the force \( F \) on the object decreases. Car airbags, crash helmets and bent knees when a person lands all extend the collision time and so reduce the harmful force.
Example: A ball of \( 0{,}15 \) kg strikes a wall at \( 8 \) \( \text{m·s}^{-1} \) and bounces back at \( 6 \) \( \text{m·s}^{-1} \). Take the initial direction as positive. The change in momentum is \( \Delta p = m(v_f - v_i) = 0{,}15(-6 - 8) = -2{,}1 \) \( \text{kg·m·s}^{-1} \). The magnitude is therefore \( 2{,}1 \) \( \text{kg·m·s}^{-1} \), and the minus indicates that the impulse is in the opposite direction to the initial motion.
1.3 Conservation of momentum
In an isolated system, where no net external force acts, the total momentum stays constant. This principle is known as the law of conservation of linear momentum. For a collision between two objects:
\( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \)
where \( u \) is the velocities before the collision and \( v \) is the velocities after the collision. We distinguish between elastic collisions, where kinetic energy is conserved, and inelastic collisions, where part of the kinetic energy is converted into other forms such as heat, sound and deformation. In a completely inelastic collision the objects stick together after the collision and move together as one. In all collisions, elastic or inelastic, momentum is however always conserved as long as the system is isolated.
This engineering application is visible everywhere: from the recoil of a gun, to the thrust of a rocket that expels gases backwards, to the way billiard balls collide with each other.