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. \(R=4H\cot \theta \)