U6: Momentum

Momentum is defined as the product of a system’s mass multiplied by its velocity

● 𝑝 = 𝑚𝑣

○ The greater an object’s mass or velocity, the greater its momentum

○ Momentum is a vector having the same direction as the velocity

Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes

● 𝐹net = 𝛥𝑝/𝛥𝑡

● It can be applied to systems where the mass is changing as well as to systems of constant mass

Impulse is the change in momentum

● 𝛥𝑝 = 𝐹net𝛥𝑡

○ A small force could cause the same impulse as a large force, but it would have to act for a much longer time

● Our definition of impulse includes an assumption that the force is constant over the time interval

○ However, forces are usually not constant

○ It is possible to find an average effective force that produces the same result as the the corresponding time-varying force

The conservation of momentum principle states that when the net external force is zero, the total momentum of the system is conserved or constant

● 𝑝tot = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

● 𝑝tot = 𝑝′tot(isolated system)

○ An isolated system is defined to be one in which the net external force is zero

An elastic collision is one that also conserves internal kinetic energy

● Internal kinetic energy is the sum of the kinetic energies of the objects in the system

An inelastic collision is one in which the internal kinetic energy changes (it is not conserved)

● A collision in which the objects stick together is called “perfectly inelastic collision”

○ It reduces internal kinetic energy more than does any other type of elastic collision

Two-dimensional collisions might cause objects to rotate before or after the collision

● To avoid rotation, we consider only the scattering of point masses - structureless particles that cannot rotate or spin

○ The conservation of momentum for the x-axis is 𝑝1x + 𝑝2x = 𝑝′1x + 𝑝′2x

■ This then leads to 𝑚1𝑣1 = 𝑚1𝑣′1 𝑐𝑜𝑠𝜃1 + 𝑚2𝑣′2 𝑐𝑜𝑠𝜃2

○ The conservation of momentum for the y-axis is 𝑝1y + 𝑝2y = 𝑝′1y + 𝑝′2y

■ This then leads to 0 = 𝑚1𝑣′1 𝑠𝑖𝑛𝜃1 + 𝑚2𝑣′2 𝑠𝑖𝑛𝜃2