Kinematics with drag

2024-09-2421:06 Status:PHYS106 Tags: Kinematics

Alternative Definitions of Newton’s Second Law

Newton’s Second Law: the rate of change of momentum is equal to the net force on that object: $\frac{d\vec{p}}{dt}={\vec{F}}_{net}$. A more familiar version would be: $\overrightarrow{F_{net}}=m\vec{a}$. However, the most useful formulation is: $\frac{d\vec{v}}{dt}=\frac{{\vec{F}}_{net}}{m}$ Newton’s 2nd law: $\frac{d\vec{v}}{dt}=\frac{1}{m}{\vec{F}}_{net}$ Where we must calculate this from $\vec{x},\vec{v}$ and knowledge of the environment. This gives us the definition of velocity (by cancelling $m$): $\frac{d\vec{x}}{dt}=\vec{v}$. In other words,

• The rate of change of position is equal to the velocity
• The rate of change of velocity is equal to the force over mass (force can be calculated from position, velocity and knowledge of the environment) (This is true for each component)

In a small amount of time $\delta t$:

• Position changes by $\vec{v}\delta t$ where $\vec{v}$ is the rate of change of position: ${\vec{x}}_{new}={\vec{x}}_{old}+v\delta t$
• Velocity changes by $\frac{\vec{F}}{m}\delta t$ where $\frac{\vec{F}}{m}$ is the rate of change of velocity ${\vec{v}}_{new}={\vec{v}}_{old}+\frac{\vec{F}}{m}\delta t$ We can predict the future $\vec{x},\vec{v}$ from present $\vec{x},\vec{v}$:

Predicting the Future

Example: Projectile Motion with Air Drag (Air Resistance)

https://www.youtube.com/watch?v=pW0JfEBE9h8 = “Dynamics and Control of Flow around Circular Cylinder” - This is very complicated so we will use the following approximation (objects of low velocity, size, + depends on viscosity - Reynold’s number).

For small velocities, viscous fluids, and small objects use: $|F_{drag}\approx bv|$ For large velocities, less viscous fluids, and large objects use: $|{\vec{F}}_{drag}\approx c{v}^2|$ Where $b$ and $c$ are numbers: measured experimentally and $v$ is the velocity

Time	Position x	Velocity v	Force F
0 s	2 m	3 m/s	- 1.5 N
0.1 s	2.3 m	2.95 m/s (3m/s + (-1.5N / 3kg) * 0.1s)	////////
0.2 s	…
$x_{new}=x_{old}+v\delta t{v}_{new}=v_{old}+\frac{F}{m}\delta t$
Doing this process by hand is very laborious, so computers often do this,
https://www.youtube.com/watch?v=GMmNKUlXXDs&t=1s