Motion Equations
Velocity
Average Velocity
$$ \vec{v}=\frac{\Delta \vec{r}}{\Delta t} $$$$
\vec{v}=\frac{\Delta \vec{r}}{\Delta t}
$$
Instant Velocity
$$ \vec{v}=\frac{d\vec{r}}{dt} $$$$
\vec{v}=\frac{d\vec{r}}{dt}
$$
Acceleration
Average Acceleration
$$ \vec{a}=\frac{\Delta \vec{v}}{\Delta t} $$$$
\vec{a}=\frac{\Delta \vec{v}}{\Delta t}
$$
Instant Acceleration
$$ \vec{a}=\frac{d\vec{v}}{dt} $$$$
\vec{a}=\frac{d\vec{v}}{dt}
$$
Non-Uniform Motion
Velocity
$$ v=\frac{ds}{dt} $$$$
v=\frac{ds}{dt}
$$
$$
\int_{s_i}^{s_f}ds=\int_{t_i}^{t_f}vdt
$$
$$
s_f-s_i=\Delta s=\int_{t_i}^{t_f}vdt
$$
$$
s_f=s_it\int_{t_i}^{t_f}vdt
$$
Acceleration
$$ a=\frac{dv}{dt} $$$$
a=\frac{dv}{dt}
$$
$$
\int_{t_i}^{t_f}adt=\int_{v_i}^{v_f}dv
$$
$$
v_f-v_i=\int_{t_i}^{t_f}adt
$$
$$
v_f=v_i+\int_{t_i}^{t_f}adt
$$
Vectors
$$ \vec{A}=\vec{A_x}+\vec{A_y} $$$$
\vec{A}=\vec{A_x}+\vec{A_y}
$$
Unit Vectors
$$ \hat{i}\parallel x $$ $$ \hat{j}\parallel y $$$$
\hat{i}\parallel x
$$
$$
\hat{j}\parallel y
$$
$$
\vec{A_x}=A_x\hat{i},
$$
$$
\vec{A_y}=A_y\hat{j}
$$
$$
\vec{A}=A_x\vec{i}+A_y\vec{j}
$$
Multiplication
Scaler
$$ \beta\vec{A}=\beta(A_x\hat{i}+A_y\hat{j}) $$$$
\beta\vec{A}=\beta(A_x\hat{i}+A_y\hat{j})
$$
Dot Product
$$ \vec{A}\cdot\vec{B}= $$Cross Product
$$ \vec{A}\times\vec{B}=\vec{C} $$$$
\vec{A}\times\vec{B}=\vec{C}
$$
$$
|\vec{A}\times\vec{B}|=|\vec{A}||\vec{B}|sin(\alpha)=|\vec{C}|
$$
\(\odot\) : out of plane \(\otimes\) : into plane
$$ if \vec{A}\parallel\vec{B} then \vec{C}=0 $$cross product is always parallel to the two vectors.
Kinematics (2D)
Trajectory
Two deminsional: x&y motions(vertical and horizontal)
Smooth line graph-looks like an arc.
Postion(vec) vs. Time(Scaler).
$$ \vec{r}=x\hat{i}+y\hat{j} $$velocity
$$ \vec{v}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j} $$Direction of Velocity?
e.g.
$$ \vec{v}=6\hat{i}+3\hat{j} $$Would be in the 1st quadrant.
$$ tan(\theta)=\frac{1}{2} $$so
$$ \theta = tan^-1 \frac{1}{2} $$slope of the trajectory arc (direction tangent to trajectory curve)
$$ tan(\theta)=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx} $$Projectile motion
(No force other than gravity)
Treat X & Y motions seperately.
how fast it moves in the x direction has no effect on motion in Y direction, and vice versa.
Y motion determines the time duration of the projectile motion.
X motion: constant velocity
Y motion: free fall
Projectile Motion PT.2
A ball(B) is launched with initial speed of \(V_o\) the launch of angle \(\theta\) can be varied. Determine \(\theta\) when l B is the largest
x-motion: uniform motion
$$ l=v_x, \Delta t $$\(\Delta t\): time of travel; \(V_x\) is a constant.
$$ V_x=V_0cos\theta $$ $$ l=V_0cos\theta\Delta t $$If we can write \(l=f(\theta)\):
$$ \frac{df(\theta)}{d\theta}=0 $$ $$ f'(\theta)=0-> $$solve for
$$ \theta $$Goal: to express \(\Delta t\) as a function of \(\theta\).
y-motion: Constant acceleration.(force of gravity)
$$ a_y=a_g=9.8m/s^2 $$needed equations:
$$ V_(yf)=V_(yi)+a_y\Delta t $$ $$ V_(yf^2)+2a_y\Delta y $$ $$ \Delta y = V_yi \Delta t t \frac{1}{2}a_y \Delta t^2 $$Use the third equation..
Relative Motion
We discuss only uniform motions here.
eg. from c’s perspective speed of B=1m/s From A’s perspective speed of B? (3m/s) speed of C=2m/s
Perspective = Reference Frame
Nothing can move faster than the speed of light. SO keeping it low speed.
Reference Frame
Coordinate system where postion and velocity are measured.
Earth is the most important reference frame.
\(s'\) is another reference frame(e.g bus, boat etc.)