Lecture 6: Momentum, Collisions, Impulse, and Energy
Momentum
Definition: Momentum is a measure of the mass in motion of an object. It is a vector quantity, meaning it has both magnitude and direction.
Calculation: The momentum (p) of a moving object is calculated as the product of its mass (m) and its velocity (v).
Formula: \vec{p} = m \vec{v}
Units: kilogram-meters per second (kg \cdot m/s)
Conservation of Momentum
Meaning: The principle of conservation of momentum states that if a system is isolated (i.e., no external net forces act on it), then the total momentum of the system remains constant over time.
In simpler terms, the total momentum before an interaction (like a collision or an explosion) is equal to the total momentum after the interaction.
Formula for a two-object system: m1\vec{v}{1,initial} + m2\vec{v}{2,initial} = m1\vec{v}{1,final} + m2\vec{v}{2,final}
Conditions for Conservation: Momentum is conserved under the condition that the system is isolated, meaning there are no external forces (or the net external force is zero) acting on it. Internal forces within the system (like forces between colliding objects) do not change the total momentum of the system.
Kinetic Energy vs. Momentum
Kinetic Energy (KE):
Definition: Kinetic energy is the energy an object possesses due to its motion. It is a scalar quantity, meaning it only has magnitude.
Calculation: KE = \frac{1}{2}mv^2
Units: Joules (J)
Momentum (p):
Definition: As defined above, momentum is the mass in motion of an object. It is a vector quantity.
Calculation: \vec{p} = m\vec{v}
Units: kilogram-meters per second (kg \cdot m/s)
How they are different:
Scalar vs. Vector: Kinetic energy is a scalar (magnitude only), while momentum is a vector (magnitude and direction).
Dependence on velocity: Kinetic energy depends on the square of the velocity (v^2), making it always positive. Momentum depends linearly on velocity (v), so its direction is the same as the velocity vector.
Conservation: Momentum is conserved in isolated systems, even in inelastic collisions. Kinetic energy is only conserved in perfectly elastic collisions.
How they are related: Both quantities depend on an object's mass and speed. They describe different aspects of an object's motion and energy state. One can be expressed in terms of the other: KE = \frac{p^2}{2m} or p = \sqrt{2mKE}.
Collisions
During a collision:
Momentum: The total momentum of the system of colliding cars (if considered an isolated system) tends to be conserved. This means the total momentum immediately before the collision equals the total momentum immediately after the collision.
Kinetic Energy: The total kinetic energy of the cars is generally not conserved. In most real-world collisions (inelastic collisions), kinetic energy is lost. It is converted into other forms of energy such as heat (due to friction and deformation), sound, and energy causing permanent deformation of the vehicles.
During an explosion:
Momentum: The total momentum of the pieces flying apart is conserved. Before the explosion, if the object was at rest, its total momentum was zero. After the explosion, the vector sum of the momenta of all the pieces will still be zero.
Kinetic Energy: The total kinetic energy of the pieces increases. The chemical potential energy stored in the explosive material is converted into kinetic energy of the fragments and other forms of energy (heat, light, sound).
Elastic Collisions:
Definition: An elastic collision is a type of collision in which both momentum and kinetic energy are conserved.
How to recognize it: In a perfectly elastic collision, objects bounce off each other without any loss of kinetic energy to heat, sound, or deformation. This is often an idealization, but billiard ball collisions can approximate elastic collisions.
Specific scenarios in one dimension:
Smaller object collides with a larger object (target at rest): The smaller object often rebounds with a reduced speed, and the larger object moves forward with a small speed.
Larger object collides with a smaller object (target at rest): The larger object continues to move in the same direction but with a slightly reduced speed, while the smaller object moves forward at a significantly higher speed (potentially almost twice the initial speed of the larger object).
Object of equal mass collides with an object (target at rest): The first object comes to a complete stop, and the second object moves forward with the exact velocity that the first object had initially. This is a classic example of momentum and kinetic energy transfer.
Recoil
Definition: Recoil is the backward movement of an object caused by the reaction force from a propelled mass. A classic example is a gun recoiling backward when a bullet is fired forward.
Determining Recoil Speed: Recoil speed can be determined using the principle of conservation of momentum. If the system (e.g., gun and bullet) is isolated and initially at rest, its total initial momentum is zero. After the action (firing), the sum of the momentum of the bullet and the momentum of the gun must still be zero. This means their momenta are equal in magnitude but opposite in direction.
Example: For a gun (mg) and bullet (mb), if initially at rest: 0 = mb vb + mg vg. Therefore, mg vg = -mb vb. The recoil velocity (vg) can be calculated as vg = -\frac{mb}{mg} v_b.
Impulse
Definition: Impulse (J) is the change in momentum of an object. It is also defined as the product of the average force (\vec{F}_{avg}) acting on an object and the time interval (\Delta t) over which the force acts.
Calculating Impulse:
From force and time: \vec{J} = \vec{F}_{avg} \Delta t
Units: Newton-seconds (N \cdot s) or kilogram-meters per second (kg \cdot m/s).
Calculating Time: If the impulse and force are known, the time it takes for a force to deliver an impulse can be calculated as: \Delta t = \frac{\vec{J}}{\vec{F}_{avg}}
Relationship to Momentum (Impulse-Momentum Theorem): Impulse is directly equal to the change in momentum of an object.
\vec{J} = \Delta \vec{p} = \vec{p}{final} - \vec{p}{initial} = m\vec{v}{final} - m\vec{v}{initial}
This theorem connects force and time with the change in an object's motion (momentum).
Airbag protection during an auto collision: An airbag works by significantly increasing the time (\Delta t) it takes for your body to stop moving during a collision. According to the impulse-momentum theorem (\vec{F}{avg} = \frac{\Delta \vec{p}}{\Delta t}), to achieve a certain change in momentum (\Delta \vec{p}), if you increase the time over which that change occurs (\Delta t), the average force (\vec{F}{avg}) exerted on your body is substantially reduced. This reduced force minimizes injury by spreading the impact over a longer duration and a larger area.
Forces and Potential Energy
Forces naturally pull things in the direction that the potential energy decreases.
Objects tend to move from higher potential energy configurations to lower potential energy configurations. For example, gravity pulls objects downwards, which is the direction where gravitational potential energy decreases. A compressed spring pushes outwards, which is the direction that decreases the elastic potential energy stored in the spring.