10-01-25 Changing Momentum

Changing Momentum

  • When momentum is changing, the instantaneous version of the Momentum Principle is often the most useful:

    • F→net=dp→dt\overrightarrow F_{net} = \frac{d \overrightarrow p}{dt}

  • If the speed is slow and the mass is constant, we will sometimes find it convenient to use acceleration:

    • F→net=dp→dt→F→net=mdv→dt→\overrightarrow F_{net} = \frac{d \overrightarrow p}{dt} \rightarrow \overrightarrow F_{net} = m \frac{d \overrightarrow v}{dt} \rightarrow

    • F→netm=dv→dt→F→netm=a→\frac{\overrightarrow F_{net}}{m} = \frac{d\overrightarrow v}{dt} \rightarrow \frac{\overrightarrow F_{net}}{m} = \overrightarrow a

1-D Coupled Systems

  • When objects move together in the same direction, their displacements are the same. This leads to their velocities and accelerations to be the same, but not their momentum.

  • Example:

Δr→1Δt=Δr→2Δt→dr→1dt=dr→2dt→v→1=v→2→dv→2dt=dv→2dt→a→1=a→2→F→net,1m1=F→net,2m2\frac{\Delta \overrightarrow r_1}{\Delta t} = \frac{\Delta \overrightarrow r_2}{\Delta t} \rightarrow \frac{d \overrightarrow r_1}{dt} = \frac{d \overrightarrow r_2}{dt} \rightarrow \overrightarrow v_1 = \overrightarrow v_2 \rightarrow \frac{d \overrightarrow v_2}{dt}=\frac{d\overrightarrow v_2}{dt} \rightarrow \overrightarrow a_1 = \overrightarrow a_2 \rightarrow \frac{\overrightarrow F_{net,1}}{m_1} = \frac{\overrightarrow F_{net,2}}{m_2}