PHYS 35100 - FALL 2024

Homework Set 1 - Solutions

  1. PDF of the solution is here.

  2. F=ma & F=bv Therefore, we can write: mdvdt=bv or using the dot notation: mx=bx

    Now we have an equation of motion for the bacterium. We have to now solve this to obtain the various kinematic quantities like position, velocity, and acceleration (as functions of time)

    Re-arrange: mdvv=bdt Now we can integrate both sides: v0vdvv=bm0tdt This is directly integrable and will lead to: lnvlnv0=bmt or using the identity lnalnb=ln(ab) ln(vv0)=bmt and solving for v(t) v(t)=v0ebmt Acceleration is just the first time derivative of velocity so we can quickly obtain: a=dvdt=v=v0m/bebmt or, by defining the time constant: τ=m/b a=v0τetτ

    Lastly, we need the position as a function of time as well. For that, we can integrate the velocity function since: v=dxdtdx=vdtdx=vdt

    In this case, v(t) is straightforward to integrate so we can see that: x(t)=0tv(t)dt=v00tetτdt Evaluating the definite integral then yields the following for the position: x(t)=v0τ(1et/τ)

    PDF of the plotting part is here.

  3. PDF of the solution is here.

    Note that the point was to actually "Work out the Taylor Series" and not just copy it down from somewhere. Please take the time to make sure you can actually obtain these series.

    The plots should look something like this.

    Here is a Colab Notebook