1 (Inertial) frame of ref

• Motion/Force/Displacement is determined wrt to observer
• If Observer is not moving or is moving at constant velocity, this is an inertial frame
• Examples
  • Inertial frame: Subway at constant velocity, you can throw a ball and catch it normally
  • Non-Inertial frame: Subway accelerating, you throw a ball but it seems to move backwards.

2 Newton’s Second Law

Solving the differential equation F=ma
Solving for 1) distance function and 2) velocity function, each wrt time

\[m \frac{ds^2}{d^2t} = f\]

mass multiply by 2nd derivative of distance function s(t) wrt to time is force.
Solve for the 2nd derivative gives us a closed form equation for the distance function s(t).
The solution allows us to predict distance traveled given time,mass and force.

DSolve[m*s''[t] == f, s[t], t] // TeXForm

\[\left\{\left\{s(t)\to \frac{f}{m}\frac{1}{2}t^2+c_2 t+c_1\right\}\right\}\] looks familiar \(s = \frac{1}{2} at^2 + v_0t\)

\[m \frac{dv}{dt} = f\]

mass multiply by 1st derivative of velocity function v(t) wrt to time is force.
Solve for the 1st derivative gives us a closed form equation for the velocity function v(t).
This solution allows us to predict velocity traveled given time, mass and force.

DSolve[m*v'[t] == f, v[t], t] // TeXForm

\[\left\{\left\{v(t)\to \frac{f t}{m}+c_1\right\}\right\}\] looks familiar \(v_n=at+v_0\)

3 Volume 1

4 Volume 3