1. $Speed=\frac{distance\ (x)}{time\ (t)}$
2. $Average\ speed =\frac{Total\ distance}{Total\ time}$
3. $$Instantaneous\ speed = \lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}$$
4. $Velocity\ v=\frac{displacement}{time}=\frac{\underset{\Delta r}{\rightarrow}}{\Delta t}$
5. $$Instantaneous\ velocity\ \vec{v}= \lim_{\Delta t\rightarrow 0}\frac{\Delta r}{\Delta t}=\frac{\underset{\Delta r}{\rightarrow}}{dt}$$
6. $Average\ acceleration\ \vec{a}= \frac{\Delta v}{\Delta t}$
7. $$Average\ acceleration\ \vec{a}= \lim_{\Delta t\rightarrow 0}\frac{\Delta v}{\Delta t}=\frac{\underset{\Delta v}{\rightarrow}}{dt}$$
8. Equation for Uniformally accelerated motion
   a. $v=v_{0}+at$
   b. $S=\left ( \frac{v_{0}+v}{2} \right )t$
   c. $S=v_{0}t+\frac{1}{2}at^{2}$
   d. $v^{2}=v_{0}^{2}+2aS$
9. Distance covered in $n^{th}$ Second $S_{n}=v_{0}+\frac{a}{2}(2n-1)$

About Vectors

A. Dot product -

$\vec{A}.\vec{B}=AB\cos \theta$

$\vec{A}.\vec{A}=\left | \vec{A} \right |^{2}$

$\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1$

$\hat{i}.\hat{j}=\hat{j}.\hat{k}=\hat{k}.\hat{i}=0$

$\cos \theta =\frac{\underset{A}{\rightarrow}.\underset{B}{\rightarrow}}{AB}$

$\vec{A}\perp \vec{B}\ then\ \vec{A}.\vec{B}=0$

$\vec{A} \parallel \vec{B}\ then\ \vec{A}.\vec{B}=AB$

 

B. Corss product -

$\vec{A}\times \vec{B}=AB\sin \theta \hat{n}$

$\vec{A}\times \vec{A}=0$

$\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=0$

$\hat{i}\times \hat{j}=\hat{k}, \hat{j}\times \hat{k}=\hat{i}, \hat{k}\times \hat{i}=\hat{j}$

$\vec{A}\times \vec{B}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\  Ax & Ay & Az\\  Bx & By & Bz \end{vmatrix}$

$\vec{A}\perp \vec{B}\ then\ \left | \vec{A}\times \vec{B} \right |=AB$

$\vec{A} \parallel \vec{B}\ then\ \left | \vec{A}\times \vec{B} \right |=0$

If $\theta$ be the angle between $\vec{A}$ and $\vec{B}$ and $\left | \vec{A} \right | = \left | \vec{B} \right |$ then

1. If $\theta = 0$ then $\left | \vec{A}+\vec{B} \right |=2A$
2. If $\theta = 180$ then $\left | \vec{A}+\vec{B} \right |=0$
3. If $\theta = 90$ then $\left | \vec{A}+\vec{B} \right |=\sqrt{2}A$
4. If $\theta = 60$ then $\left | \vec{A}+\vec{B} \right |=\sqrt{3}A$
5. If $\theta = 120$ then $\left | \vec{A}+\vec{B} \right |=A$

 

For Projectile

1. Time to reach the highest point $t_{max}=\frac{v_{0}\sin \theta }{g}$ 
2. Maximum height $H=\frac{v_{0}^{2}\sin ^{2}\theta }{2g}$ 
3. Range $R=\frac{v_{0}^{2}\sin 2\theta }{g}$ 
4. Maximum Range $R=\frac{v_{0}^{2}}{g}$ 
5. Flight time $T=\frac{2v_{0}\sin \theta }{g}$ 
6. Equation of trajectory $y=x\tan \theta -\frac{gx^{2}}{2v_{0}^{2}\cos ^{2}\theta }$ 
7. $R=4H\cot \theta$