physics-class-11-unit-04-chapter-03-planar-motion-motion-in-a-plane-lecture-3-4-gxd4tqd5kgu

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Planar motion Motion in a Plane </primary>
<hr>
<main>

- Let us say a P particle is moving along some path.

- We are studying this with respect to some reference frame in  which we have fixed coordinate axis.

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Planar motion Motion in a Plane $\rightarrow$ <span style = "color:blue"> Planar motion Motion in a Plane </span> $\rightarrow$ Planar motion Motion in a Plane $\rightarrow$ Average velocity </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Planar motion Motion in a Plane </primary>
<hr>
<main>

- $ \overrightarrow{p p^{\prime}}  =\Delta \vec{r}$ =$\vec{r}(t+\Delta t)-\vec{r}(t) $

- $\overrightarrow{v_p}= \text{limit}_{\Delta t \rightarrow 0} \frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$

- $\vec{r}(t+\Delta t)= (t+\Delta x) \hat{\imath}+(y+\Delta y) \hat{\jmath} $

- $\vec{r}(t)=x \hat{\imath}+y \hat{\jmath}$

- $\Delta \vec{r}=  \Delta x \hat{\imath}+\Delta y \hat{\jmath} $

- $\vec{V}$ =$\frac{\Delta x}{\Delta t} \hat{\imath}$ + $\frac{\Delta y}{\Delta t} \hat{\jmath} =v_2 \hat{\imath}+v_t \hat{\jmath}$

</div>
	
<div style='float: right; width:0%;'>
	
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-04-chapter-03-planar-motion-motion-in-a-plane-lecture-3_4-gxd4tqd5kgu-064-0297.6.jpg)

</main>
<hr>
<footer style = "font-size: 18px">Planar motion Motion in a Plane $\rightarrow$ Planar motion Motion in a Plane $\rightarrow$ <span style = "color:blue"> Planar motion Motion in a Plane </span> $\rightarrow$ Average velocity $\rightarrow$ Magnitude of Vector </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Average velocity </primary>
<hr>
<main>

* $\Delta t \rightarrow$ finite interval
* $\overrightarrow{\vec{v}}=\frac{\overrightarrow{\Delta r}}{\Delta t}$
* Direction of $\vec{v}$ as direction of $\Delta \vec{r}$
* In limit $\Delta t \rightarrow 0$
* Direction of $\overrightarrow{\Delta r}$ is tangent to the path
* Direction of $\vec{v}$ is always tangent to the path

</div>

</main>
<hr>
<footer style = "font-size: 18px">Planar motion Motion in a Plane $\rightarrow$ Planar motion Motion in a Plane $\rightarrow$ <span style = "color:blue"> Average velocity </span> $\rightarrow$ Magnitude of Vector $\rightarrow$ Relative velocity in 2D motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Acceleration </primary>
<hr>
<main>

- Acceleration $=\frac{\Delta \vec{v}}{\Delta t}$.

- The direction of acceleration has to be along $\Delta \vec{v}$.

</div>
	
<div style='float: right; width:50%;max-width:450px;'>

</main>
<hr>
<footer style = "font-size: 18px">Acceleration $\rightarrow$ Acceleration $\rightarrow$ <span style = "color:blue"> Acceleration </span> $\rightarrow$ Acceleration on a curved path $\rightarrow$ Acceleration on a curved path </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Acceleration on a curved path </primary>
<hr>
<main>

- Accelaration on a curved path is due to two components.

- a) change in speed of the particle along $\hat{e}_t$

- b) $2^{\text {nd }}$ component is perpendicular to the $\hat{e_t}$, because direction of $\vec{v}$ changes due to the curved nature of path.
	
</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2a8f8f5c-1539-4eed-6520-835bebc08200/public)

</main>
<hr>
<footer style = "font-size: 18px">Acceleration $\rightarrow$ Acceleration $\rightarrow$ <span style = "color:blue"> Acceleration on a curved path </span> $\rightarrow$ Acceleration on a curved path $\rightarrow$ Acceleration on a curved path </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Acceleration on a curved path </primary>
<hr>
<main>

- Even if the magnitudes of v and v' are equal, but because the directions are different, that will lead to a component of $\Delta \vec{v}$ perpendicular to the $\hat{e_t}$ direction.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Acceleration $\rightarrow$ Acceleration on a curved path $\rightarrow$ <span style = "color:blue"> Acceleration on a curved path </span> $\rightarrow$ Acceleration on a curved path $\rightarrow$ Acceleration on a curved path </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Acceleration on a curved path </primary>
<hr>
<main>

- $2^{\text {nd }}$ component perpendicular to the $\hat{e_t}$, points towards the centre of circle in which the particle is moving.

- $a_n \propto (\text { speed })^2$

- If locally, the particle is moving in a circle of radius $R$.

</div>
<div style='float: right; width:50%;max-width:450px;'>

</main>
<hr>
<footer style = "font-size: 18px">Acceleration on a curved path $\rightarrow$ Acceleration on a curved path $\rightarrow$ <span style = "color:blue"> Acceleration on a curved path </span> $\rightarrow$ Acceleration on a curved path $\rightarrow$ Uniform Circular Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Uniform Circular Motion </primary>
<hr>
<main>

- Particle following a circular path.

- Uniform $\Rightarrow$ constant speed $=v$

- $\vec{a_p}=\frac{v^2}{R}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Acceleration on a curved path $\rightarrow$ Acceleration on a curved path $\rightarrow$ <span style = "color:blue"> Uniform Circular Motion </span> $\rightarrow$ Uniform Circular Motion $\rightarrow$ Uniform Circular Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Uniform Circular Motion </primary>
<hr>
<main>

- $\Delta \theta \rightarrow$ angle covered by the particle with respect to the centre of circle,

- $\Delta \theta \rightarrow$ angular displacement

- $\Delta t \rightarrow$ time taken for particle to go from $P \rightarrow P^{\prime}$

</div>
	
<div style='float: right; width:50%;'>
	
![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b93fea9f-0990-497c-4427-29bfe3f3ad00/public)

</main>
<hr>
<footer style = "font-size: 18px">Acceleration on a curved path $\rightarrow$ Uniform Circular Motion $\rightarrow$ <span style = "color:blue"> Uniform Circular Motion </span> $\rightarrow$ Uniform Circular Motion $\rightarrow$ Uniform Circular Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Uniform Circular Motion </primary>
<hr>
<main>

- Angular velocity =$\frac{\Delta \theta}{\Delta t}=\omega$

- $\omega=\frac{\Delta \theta}{\Delta t}$

- $ v=\frac{PP^{\prime}}{\Delta t}$

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-04-chapter-03-planar-motion-motion-in-a-plane-lecture-3_4-gxd4tqd5kgu-276-1423.2.jpg)

</main>
<hr>
<footer style = "font-size: 18px">Uniform Circular Motion $\rightarrow$ Uniform Circular Motion $\rightarrow$ <span style = "color:blue"> Uniform Circular Motion </span> $\rightarrow$ Uniform Circular Motion $\rightarrow$ Uniform Circular Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Uniform Circular Motion </primary>
<hr>
<main>

- speed = $\frac{\Delta S}{\Delta t}$=$\frac{\widehat{P P^{\prime}}}{\Delta t}$=$\frac{R \Delta \theta}{\Delta t}$

- =R$ \omega $

- acceleration =$\frac{v^2}{R}$=$\frac{R^2 \omega^2}{R}$=R$\omega^2$

- acceleration towards centre $\rightarrow$ <span style="color:blue">centripetal acceleration</span>.

</div>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Uniform Circular Motion </primary>
<hr>
<main>

- $ \omega=\frac{2 \pi}{T}=2 \pi \nu $

- $ v=R \omega=2 \pi R \nu $

- $ a=R \omega^2=4 \pi^2 \nu^2 R$

- If circular motion is not uniform speed, the speed of the particle changes.

- $\Rightarrow \omega$ is not constant.

- $\alpha \rightarrow$ angular acceleration =$\frac{d \omega}{d t}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Uniform Circular Motion $\rightarrow$ Uniform Circular Motion $\rightarrow$ <span style = "color:blue"> Uniform Circular Motion </span> $\rightarrow$ For General Circular Motion $\rightarrow$ Constant Acceleration </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Constant Acceleration </primary>
<hr>
<main>

- $ \vec{a}=\frac{\vec{v}-\vec{v}_ 0}{(t-0)}=\frac{\vec{v}-\vec{v}_0}{t} $

- $ \vec{v}=\overrightarrow{v_ 0}+\overrightarrow{a_0 t} $

- $ v_ x=v_ {0 x} + a_x t $

- $ v_ y=v_ 0 y + a_y t $

- $ \vec{a}$ = $a_ x \hat{\imath}+a_y \hat{\jmath}$

- $ \vec{v}$ = $v_x \imath+v _ y \jmath$

- $\overrightarrow{v_ 0}$ = $v_ {0x} \hat{\imath}$+  $v_{0y}\hat{\jmath}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">For General Circular Motion $\rightarrow$ Constant Acceleration $\rightarrow$ <span style = "color:blue"> Constant Acceleration </span> $\rightarrow$ Constant Acceleration $\rightarrow$ Constant Acceleration </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Constant Acceleration </primary>
<hr>
<main>

- $\vec{r}$ (position vector) at time $t$

- given $\vec{r}_0$ at $t=0$.

- Average velocity =$\frac{\overrightarrow{v_0}+\vec{v}}{2}$

- displacement =$\vec{r}-\overrightarrow{r_0}$= $(\frac{\overrightarrow{v_0}+\vec{v}} {2}) t $

- =$\frac{\overrightarrow{v_0}+(\overrightarrow{v_0}+\vec{a} t) t}{2}$

</div>
	
<div style='float: right; width:0%;'>

</main>
<hr>
<footer style = "font-size: 18px">Constant Acceleration $\rightarrow$ Constant Acceleration $\rightarrow$ <span style = "color:blue"> Constant Acceleration </span> $\rightarrow$ Constant Acceleration $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Constant Acceleration </primary>
<hr>
<main>

- $\vec{r}-\vec{r}_0=\vec{v} _0 t+\frac{1}{2} \vec{a} t^2 $

- $ \vec{r}=\vec{r}_0+\vec{v} _0 t+\frac{1}{2} \vec{a} t^2 $
 
- $ x=x_0+v _ {0 x} t+\frac{1}{2} a _ x t^2 $

- $ y=y_0+v _ 0 y t+\frac{1}{2} a _y t^2 $

- Equation of the path: relate x to y.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Constant Acceleration $\rightarrow$ Constant Acceleration $\rightarrow$ <span style = "color:blue"> Constant Acceleration </span> $\rightarrow$ Projectile Motion $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
<hr>
<main>

* **Motion of a particle which is travelling under the influence of gravity**
* **Motion of a cricket ball after it has been hit by the bat**
* Projectile can have an $x$ component of velocity at $t=0$
* Upward direction as +y
* $x \rightarrow$ horizontal direction
* $\vec{a}=0 \hat{\imath}+-g \hat{\jmath}$
* $t=0$, initial time, particle be at $(0,0)$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Constant Acceleration $\rightarrow$ Constant Acceleration $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Projectile Motion $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
<hr>
<main>

- Initial velocity $\vec{v_0}$

- The speed $v_0 $ and the angle $\theta_0$.

- $ v_{0 x}=v_0 \cos \theta_0 $

- $ v_{0 y}=v_0 \sin \theta_0 $

- We wish to find coordinate of projectile (p) at a later time.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Constant Acceleration $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Projectile Motion $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
<hr>
<main>

- $ \vec{a}=-g \hat{\jmath}, a_x=0, a_y=-g $

- $ \vec{v}_0=\left(v_0 \cos \theta_0\right) \hat{\imath}+\left(v_0 \sin \theta_0\right) \hat{\jmath} $

- $ \vec{v}_0$=(0,0) $ {x_0}=0$, $\{y_0}=0$

- x=$(v_0 t)=(v_0 \cos \theta_0) t $

- $ y=(v_0 \sin \theta_0) t-\frac{1}{2} g t^2 $

- **<span style="color:green">Eliminate t</span>**

- $ v_x=v_0 \cos \theta_0 $

- $ v_y=v_0 \sin \theta_0-g t$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Projectile Motion $\rightarrow$ Time taken for projectile to reach maximum height </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
<hr>
<main>

- t=$\frac{x}{v_0 \cos \theta_0}$

- y=$v_0 \sin \theta_0 \frac{x}{v_0 \cos \theta_0}$ -$\frac{1}{2}$ g $\frac{x^2} {v_0^2\cos^2 \theta_0}$

- =$\tan \theta_0 x-\frac{1}{2} \frac{g}{v_0^2 \cos ^2 \theta_0} x^2$
 
- =$a x+b x^2 $

- <span style="color:blue">A particle which is thrown as a projectile has the path taken by that is of a parabola.</span>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Time taken for projectile to reach maximum height $\rightarrow$ Range </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Range </primary>
<hr>
<main>

- $(v_0 \sin \theta_0)$ $\frac{v_0 \sin \theta_0}{g}$-$\frac{1}{2} g \cdot \frac{v_0^2 \sin \theta_0^2}{g^2} $

- =$\frac{v_0{ }^2 \sin ^2 \theta_0}{2 g}$

- <span style="color:blue"><ins>**Range**</ins></span>: Time taken for flight =$T_f$

- time at which $y=0$.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Time taken for projectile to reach maximum height $\rightarrow$ <span style = "color:blue"> Range </span> $\rightarrow$ Projectile Motion $\rightarrow$ Maximum Range </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
<hr>
<main>

- 0 = $v_0 \sin \theta_0 t$  - $\frac{1}{2} g t^2 $

- $ T_f$ = $\frac{2 v_0 \sin \theta_0}{g}$ 
 
- R = x = $(v_0 \cos \theta_0) T_f $

- = $(v_0 \cos \theta_0)$ $\cdot$ $\frac{2 v_0 \sin \theta_0}{g}$

- = $\frac{v_0^2 \sin \(2 \theta_0)}{g}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Time taken for projectile to reach maximum height $\rightarrow$ Range $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Maximum Range $\rightarrow$ Assumption </footer>

### <Success style="font-size:25px"> Planar Motion Motion In A Plane L-3 </Success>
### <primary style="font-size:24px"> Maximum Range </primary>
<hr>
<main>

- R= $\frac{v_0^2 \sin 2 \theta_0}{g}$

- Maximum R for a given $v_0$ occurs at $\theta_0=45^0$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Range $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Maximum Range </span> $\rightarrow$ Assumption $\rightarrow$ Thank you </footer>