SPH4U - Unit 3 - Linear Momentum

Linear Momentum

  • Momentum - The vector that represents the product of the mass and velocity of an object

    • \overrightarrow{p}=m\cdot\overrightarrow{v}

    • An object with no velocity has a momentum of zero

Impulse

  • Momentum changes only when an external force acts on the observed object

    • Momentum will be constant if no external force is present

    • This holds with Newton’s first law, because velocity can only change if there is an external force acting on it

  • Impulse - The vector that represents the change in momentum

    • \overrightarrow{J}=\overrightarrow{p_{f}}-\overrightarrow{p_{i}}

    • Impulse is also represented by the area of a force-time graph

      • \overrightarrow{J}=\overrightarrow{F_{avg}}\cdot\Delta t

  • Impulse-Momentum Theorem - The average force of an object equals the product of the mass and change in velocity of an object over time

    • This is a reiteration of Newton’s second law

Conservation of Linear Momentum

  • The total momentum can only change if it is not isolated and has external interactions

  • Explosion - Two objects are travelling together at the same velocity, and then are separated by the same force

    • The total momentum of the system is the addition of the individual momentum of each individual part

    • The velocity of the centre of mass of a system will remain constant

    • When m1v1 = m2v2, a linear relationship will be formed, which represents momentum

  • For inelastic and elastic collisions, the total momentum of the system is the addition of each individual object’s momentum

    • If momentum of one of the objects is negative, it becomes subtraction

    • The velocity of the centre of mass:

      • \overrightarrow{v}_{c.m.}=\frac{\Sigma\left(\overrightarrow{p_{i}}_{}\right)}{\Sigma\left(\overrightarrow{m_{i}}\right)}

  • If there are no external interactions, the momentum of a system is equal both before and after the collision

    • If objects start at rest, their initial momentum is zero

      • This means that the final momentum must also equal zero, which is possible when they have equal magnitudes of momentum in opposite directions

  • The slope of a momentum-time graph is force

  • The area of a force-time graph is the change in momentum

  • Momentum bar charts

    • The bars represent initial and final momentum

      • If there is an external force, the bars will not have the same sum in the initial and final chart

  • If there is an external force involved, impulse should be considered as part of the equation

    • \overrightarrow{F}\Delta t=\overrightarrow{\Delta p}=\overrightarrow{J}

Elastic and Inelastic Collisions

  • Elastic Collision - A collision where the two colliding objects “bounce” in different directions

    • Momentum is constant

    • The total kinetic energy is maintained (Ki = Kf)

  • Inelastic Collision - A collision where the two colliding objects “stick” and travel in the same direction

    • Momentum is constant

    • The total kinetic energy decreases (Ki > Kf)

  • The total kinetic energy must be calculated before and after to differentiate between the two types

    • E_{k}=m\overrightarrow{v}\frac12

Energy

  • Mechanical energy is the sum of all kinetic and potential energy

    • For potential energy:

      • U=mg\Delta y

  • Mechanical energy is conserved, so that ME_{i^{}}=ME_{i}