Honors Physics I - 2 Force and Motion

Relative Motion

Galilean transformation :

$x_{PA}=x_{PB}+x_{AB}$

Projectile motion

$x=v_0\cos\theta\, t\\ y=y_0+v_0\sin\theta\,t-\frac 1 2gt^2=0\\ R=\frac {v_0^2}g\sin2\theta_0$

‪Projectile Motion‬

Uniform Circular motion

$a=\frac {v^2}R$

*Basic Calculus

Derivative

$\frac{df(x)}{dx}=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Some examples :

$\frac{d}{dx}\ln x=\frac 1 x$

Proof :

Note that :

$e = \lim_{n\to +\infty}(1+\frac 1 n)^n$

$\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$

For $\theta \in (0,\frac{\pi}2)$ :

$\sin \theta<\theta<\tan \theta$

Therefore : $\frac d {dx}\sin x=\cos x$ (Try!)

Chain Rules

Integration

Use the partition. Fundamental Theorem of Calculus :

$\int_0^xf(t)dt=\lim_{N\to +\infty}\sum_{i=1}^Nf(t_i)\Delta t$

Force and Emotion

Newton’s first law : If no force acts on a body, the body’s velocity cannot change; that is , the body cannot accelerate (Works in inertial reference frame)
Newton’s second law : The net force on a body is equal to the product of the body’s mass and its acceleration.

Unit : N (Newton) $=\text{kg}\times\text{m/s}^2$

Internal force is the force between two objects in the system.

Remember to put the total force and direction (or the vector or even draw or figure) for you answer

Special Force

Gravitational force
Normal force
Friction and tension (a cord pulled taut)

Newton’s third law : When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction

Friction

Resistivity : 电阻 (superconductor)

static frictional force
kinetic frictional force

Why the maximum static frictional force is a bit larger than the kinetic frictional force :

When the body is static, it is more likely to track at the lower potential

$\mu_s$ is the coefficient static friction.

The Drag Force and Terminal Speed

An empirical law of fluid, the drag force is

$D=\frac 1 2C\rho Av^2$

Where :

$v$ is the relative velocity
$\rho$ is the air density (mass/volume)
$C$ is the experimentally determined drag coefficient
$A$ is the effective cross-sectional area of the body (the area taken perpendicular to the relative velocity)