Linear Momentum and Collisions Lesson 9 by Owen Borville 11.24.2025
Linear momentum is the product of a system's mass by its velocity: p = mv, where m is the mass of the system, v is the velocity. The SI unit for momentum is kg * m/s. Momentum is a vector quantity along with velocity.
Newton's second law of motion in terms of momentum states that the external force equals the change in momentum of a system divided by the time over which it changes.
Newton's Second Law of Motion: Fnet = Δp/ Δt, where Fnet is the net external force, Δp is the change in momentum, and Δt is the change time.
Impulse is the change in momentum and equals the average net external force multiplied by the time this force acts: Δp = Fnet Δt. Forces are usually not constant over a period of time.
Impulse can also be found with the integral of net force (J), a vector, from ti to tf (initial and final times): J = ∫(tf, ti) F(t)dt
The impulse-momentum theorem states: J(vector) = Δp (vector)
The average force from momentum is F(vector) = Δp(vector)/Δt
Instantaneous force from momentum (Newton's second law) F(t) (vector) = dp/dt
Conservation of momentum: p1 + p2 = constant or dp1/dt + dp2/dt = 0
Conservation of momentum is p(total) = constant or ptot = p'(tot) (isolated system), p(tot) is the initial momentum and p'(tot) is the total momentum some time later. A closed or isolated system is one in which mass is constant and the net external force is zero (Fnet = 0). The law of conservation of momentum states that a closed system is constant in time or conserved. Conservation of momentum is conserved only when the system is closed.
During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero. Conservation of momentum applies only when the net external force is zero. The conservation of momentum principle is valid when considering systems of particles.
Elastic collisions are those that conserve internal kinetic energy. Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions. Momentum is conserved whether or not kinetic energy is conserved.
Inelastic collisions are those that the internal kinetic energy changes (it is not conserved). A collision in which one object sticks together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision. Applications of momentum, rotational motion, and vibrations are used in sports science and technology.
Problems are solved using two-dimensional collisions by choosing a convenient coordinate system and breaking the motion into components along perpendicular axes. The coordinate system with the x-axis parallel to the velocity of the incoming particle should be chosen. Momentum is conserved in both directions.
Two dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (x axis): m1v1 = m1v'1cos θ1 + m2v'2cos θ2
and along the direction perpendicular to the initial direction (y-axis): 0 = m1v'1y + m2v'2y. (use Pythagorean theorem).
The inertial kinetic force before and after the collision of two objects that have equal masses:
1/2mv1^2 = 1/2mv'1^2 + 1/2mv'2^2 + mv'1v'2cos(θ1-θ2) Point masses are structureless particles that cannot spin.
The center of mass is a defined position vector of an extended object made up of many objects. The center of mass is the average location of the total mass of the object. The center of mass of an object traces out the trajectory of Newton's second law, due to the net external force. Internal forces within an extended object cannot alter the momentum of the extended object as a whole.
External forces = Fext (vector)= dpj/dt
Newton's Second Law for an extended object. F(vector) = dpCM/dt
The acceleration of center of mass(vector) is a(CM) = net external force Fext/M(total mass).
The velocity of the center of mass (vector) is the total momentum of the system divided by the total mass: vCM = mi*vi/mi or vCM = total pi/total mi
The position of center of mass (vector) = rCM = total mi*ri/mi. For continuous mass distribution: rCM = 1/M ∫r dm
A rocket is an example of conservation of momentum where the mass of the system is not constant, as the rocket ejects fuel to provide thrust.
For Rocket Propulsion, Newton's Third Law of Motion states that to every action, there is an equal and opposite reaction. Acceleration of a rocket is a = ve/m( Δm/Δt)-g.
The rocket equation gives the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass: Δv = u ln (mi/m)
A rocket's acceleration depends on three main factors: (1) the greater the exhaust velocity of the gases, the greater the acceleration. (2) the faster the rocket burns its fuel, the greater its acceleration. (3) the smaller the rocket's mass, the greater the acceleration.
Linear momentum is the product of a system's mass by its velocity: p = mv, where m is the mass of the system, v is the velocity. The SI unit for momentum is kg * m/s. Momentum is a vector quantity along with velocity.
Newton's second law of motion in terms of momentum states that the external force equals the change in momentum of a system divided by the time over which it changes.
Newton's Second Law of Motion: Fnet = Δp/ Δt, where Fnet is the net external force, Δp is the change in momentum, and Δt is the change time.
Impulse is the change in momentum and equals the average net external force multiplied by the time this force acts: Δp = Fnet Δt. Forces are usually not constant over a period of time.
Impulse can also be found with the integral of net force (J), a vector, from ti to tf (initial and final times): J = ∫(tf, ti) F(t)dt
The impulse-momentum theorem states: J(vector) = Δp (vector)
The average force from momentum is F(vector) = Δp(vector)/Δt
Instantaneous force from momentum (Newton's second law) F(t) (vector) = dp/dt
Conservation of momentum: p1 + p2 = constant or dp1/dt + dp2/dt = 0
Conservation of momentum is p(total) = constant or ptot = p'(tot) (isolated system), p(tot) is the initial momentum and p'(tot) is the total momentum some time later. A closed or isolated system is one in which mass is constant and the net external force is zero (Fnet = 0). The law of conservation of momentum states that a closed system is constant in time or conserved. Conservation of momentum is conserved only when the system is closed.
During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero. Conservation of momentum applies only when the net external force is zero. The conservation of momentum principle is valid when considering systems of particles.
Elastic collisions are those that conserve internal kinetic energy. Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions. Momentum is conserved whether or not kinetic energy is conserved.
Inelastic collisions are those that the internal kinetic energy changes (it is not conserved). A collision in which one object sticks together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision. Applications of momentum, rotational motion, and vibrations are used in sports science and technology.
Problems are solved using two-dimensional collisions by choosing a convenient coordinate system and breaking the motion into components along perpendicular axes. The coordinate system with the x-axis parallel to the velocity of the incoming particle should be chosen. Momentum is conserved in both directions.
Two dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (x axis): m1v1 = m1v'1cos θ1 + m2v'2cos θ2
and along the direction perpendicular to the initial direction (y-axis): 0 = m1v'1y + m2v'2y. (use Pythagorean theorem).
The inertial kinetic force before and after the collision of two objects that have equal masses:
1/2mv1^2 = 1/2mv'1^2 + 1/2mv'2^2 + mv'1v'2cos(θ1-θ2) Point masses are structureless particles that cannot spin.
The center of mass is a defined position vector of an extended object made up of many objects. The center of mass is the average location of the total mass of the object. The center of mass of an object traces out the trajectory of Newton's second law, due to the net external force. Internal forces within an extended object cannot alter the momentum of the extended object as a whole.
External forces = Fext (vector)= dpj/dt
Newton's Second Law for an extended object. F(vector) = dpCM/dt
The acceleration of center of mass(vector) is a(CM) = net external force Fext/M(total mass).
The velocity of the center of mass (vector) is the total momentum of the system divided by the total mass: vCM = mi*vi/mi or vCM = total pi/total mi
The position of center of mass (vector) = rCM = total mi*ri/mi. For continuous mass distribution: rCM = 1/M ∫r dm
A rocket is an example of conservation of momentum where the mass of the system is not constant, as the rocket ejects fuel to provide thrust.
For Rocket Propulsion, Newton's Third Law of Motion states that to every action, there is an equal and opposite reaction. Acceleration of a rocket is a = ve/m( Δm/Δt)-g.
The rocket equation gives the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass: Δv = u ln (mi/m)
A rocket's acceleration depends on three main factors: (1) the greater the exhaust velocity of the gases, the greater the acceleration. (2) the faster the rocket burns its fuel, the greater its acceleration. (3) the smaller the rocket's mass, the greater the acceleration.