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 $$S_{n}=v_{0}+\frac{a}{2}(2n-1)$$

A. Dot product -

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

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

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

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

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

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

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

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

$$\vec{A}\times \vec{A}=0$$

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

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

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

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

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

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. $$R=4H\cot \theta$$