physics-class-11-unit-04-chapter-04-problemsin-kinematics-motion-in-a-plane-lecture-n96pnciey5y

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
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| **Horizontal** | **Vertical** |
| -------- | -------- |
| $a_x=0$    | $a_y=- g$     |
| $v_{x0} = v_0 \cos \theta_0$    | $v_{y0}=v_0 \sin \theta_0 $     |

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<footer style = "font-size: 18px"> $\rightarrow$ Problems in Kinematics $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Projectile Motion $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
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| <span style="color:green">Horizontal</span> |<span style="color:green"> Vertical</span> |
| -------- | -------- |
| $ v_x=v_0 \cos \theta_0+a_2 t $    | $ v_y=v_0 \sin \theta_0-g t $     |
| $ x=v_0 \cos \theta_0 $    | $ y=v_0 \sin \theta_0 t-\frac{1}{2} g t^2$     |

- y as a function of x- eliminating time

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<footer style = "font-size: 18px">Problems in Kinematics $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Projectile Motion $\rightarrow$ Projectile Motion </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
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- $ t=\frac{x}{v_0 \cos \theta_0}$

- $ y=\frac{v_0 \sin \theta_0 x}{v_0 \cos \theta_0}$-$\frac{1}{2} g (\frac{x}{v_1 \cos \theta_0})^2$

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

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

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
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- <span style="color:green">Time flight = T</span>

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

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

- R = $ v_0 \cos \theta_0$ T=$\frac{v_0 \cos \theta_0 2 v_0 \sin \theta}{g} $

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

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![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-04-problemsin-kinematics-motion-in-a-plane-lecture-n96pnciey5y-102-0488.4.jpg)

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile Motion </primary>
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- $ y=v_0 \sin \theta_0 t-\frac{1}{2} g t^2$

- $\Delta t = t_2 - t_1$

- Quadratic equation in $t$

- The two roots of this quadratic equation give the value $ t_1$ and $t_2$.

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<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Projectile Motion </span> $\rightarrow$ Quadratic Equation $\rightarrow$ Maximum Height </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Quadratic Equation </primary>
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- Sum of roots = -$\frac{b}{a}$

- Product of roots =$\frac{c}{a}$

- $a x^2+b x+c=0$

- $(\alpha+\beta)^2$-4 $\alpha \beta$=$(\alpha-\beta)^2 $

- $(\alpha-\beta)^2$=$(\text { sum of roots })^2$-4 $(\text{ product of roots })$

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<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Projectile Motion $\rightarrow$ <span style = "color:blue"> Quadratic Equation </span> $\rightarrow$ Maximum Height $\rightarrow$ Maximum Height </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Maximum Height </primary>
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* $H=$  maximum, when $v_y=0$
* t=$\frac{v_0 \sin \theta_0}{g}$
* y=$v_0 \sin \theta_0 \frac{v_0 \sin \theta_0}{g}$-$\frac{1}{2}$ g $\frac{v_0^2 \sin ^2 \theta}{g^2}$
* H=$\frac{v_0^2 \sin ^2 \theta_0}{2 g}$ $\leftarrow $ maximum height attained by projectile

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![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-04-problemsin-kinematics-motion-in-a-plane-lecture-n96pnciey5y-176-0860.4.jpg)

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<footer style = "font-size: 18px">Projectile Motion $\rightarrow$ Quadratic Equation $\rightarrow$ <span style = "color:blue"> Maximum Height </span> $\rightarrow$ Maximum Height $\rightarrow$ Trajectory Formula </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Trajectory Formula </primary>
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- y (in terms of R )

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

- =x $\tan \theta_0-\frac{g}{2 v_0^2 \cos^2 \theta_0} \cdot \frac{\sin\theta_0}{\sin \theta_0}$

- =x $\tan \theta_0 - \frac{\sin \theta_0}{R \cos \theta_0}$

- =x $\tan \theta_0- \frac{g}{R} \cdot \tan \theta_0$

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<footer style = "font-size: 18px">Maximum Height $\rightarrow$ Maximum Height $\rightarrow$ <span style = "color:blue"> Trajectory Formula </span> $\rightarrow$ Trajectory Formula $\rightarrow$ Projectile on an inclined plane </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Trajectory Formula </primary>
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- y in terms of R

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

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

- = x$ \tan \theta_0\left(1-\frac{x}{R}\right) $

- = x$ \tan \theta_0 \frac{(R-x)}{R}$

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![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-04-problemsin-kinematics-motion-in-a-plane-lecture-n96pnciey5y-212-1107.6.jpg)

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<footer style = "font-size: 18px">Maximum Height $\rightarrow$ Trajectory Formula $\rightarrow$ <span style = "color:blue"> Trajectory Formula </span> $\rightarrow$ Projectile on an inclined plane $\rightarrow$ Projectile on an inclined plane </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile on an inclined plane </primary>
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- $\tan \alpha=\frac{y}{x} $

- $\tan \beta=\frac{y}{R-x} $

- $\tan \alpha+\tan \beta$=$\frac{y}{x}+\frac{y}{R-x}$

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<footer style = "font-size: 18px">Trajectory Formula $\rightarrow$ Trajectory Formula $\rightarrow$ <span style = "color:blue"> Projectile on an inclined plane </span> $\rightarrow$ Projectile on an inclined plane $\rightarrow$ Projectile on an inclined plane </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile on an inclined plane </primary>
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- $\tan \alpha + \tan \beta $ =$\frac{y(R-x)+y x}{x(R-x)}$

- $\quad =\frac{y R}{x(R-x)} $

- y = x $\tan \theta_0 \frac{(R-x)}{R}$

- $\frac{y R}{(R-x) x} $= $\tan \theta_0 $

- $\Rightarrow \tan \alpha +\tan \beta=\tan \theta_0 $

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<footer style = "font-size: 18px">Trajectory Formula $\rightarrow$ Projectile on an inclined plane $\rightarrow$ <span style = "color:blue"> Projectile on an inclined plane </span> $\rightarrow$ Projectile on an inclined plane $\rightarrow$ Projectile on an inclined plane </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile on an inclined plane </primary>
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- $ R=\frac{v_0^2 \sin (2 \theta_0)}{g} $

- $v_0, \theta_0 \rightarrow $

- $R_1 =v_0^2 \sin ^2 \frac{2 \theta_0}{g}$

- $v_0\left(\frac{\pi}{2}-\theta_0\right) \rightarrow $

- $ R_2=v_0^2 \sin ^2 \left(\frac{2\pi}{2}-2 \theta_0\right)$

- $ =R_1$

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<footer style = "font-size: 18px">Projectile on an inclined plane $\rightarrow$ Projectile on an inclined plane $\rightarrow$ <span style = "color:blue"> Projectile on an inclined plane </span> $\rightarrow$ Projectile on an inclined plane $\rightarrow$ Resolve Accelaration </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Projectile on an inclined plane </primary>
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- $v_0$

- $\theta_0 \rightarrow \angle$ from ground

- $\alpha \rightarrow \angle$ inclination

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<footer style = "font-size: 18px">Projectile on an inclined plane $\rightarrow$ Projectile on an inclined plane $\rightarrow$ <span style = "color:blue"> Projectile on an inclined plane </span> $\rightarrow$ Resolve Accelaration $\rightarrow$ Resolve Accelaration </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Resolve Accelaration </primary>
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- $a_x=-g \sin \alpha $

- $a_y=-g \cos \alpha $

- $v_{x_0}=v_0 \cos (\theta_0-\alpha) $

- $v_{y_0}=v_0 \sin (\theta_0-\alpha) $

- $v_x=v_0 \cos (\theta_0-\alpha)-g \sin \alpha t $

- $v_y=v_0 \sin \left(\theta_0-\alpha\right)-g \cos \alpha t $

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<footer style = "font-size: 18px">Projectile on an inclined plane $\rightarrow$ Resolve Accelaration $\rightarrow$ <span style = "color:blue"> Resolve Accelaration </span> $\rightarrow$ Resolve Accelaration $\rightarrow$ Trajectory Formula </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Range </primary>
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- $R_h=R\cos \alpha$

- $R_h=v_0 \cos \theta_0 T$

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

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<footer style = "font-size: 18px">Resolve Accelaration $\rightarrow$ Trajectory Formula $\rightarrow$ <span style = "color:blue"> Range </span> $\rightarrow$ Range $\rightarrow$ Parachute Problems </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Range </primary>
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- R $\cos \alpha$=$v_0 \cos \theta_0 \cdot \frac{2 v_0 \sin (\theta_0 \alpha)}{g \cos \alpha}$ 
 
- R=$\frac{2 v_0^2 \cos \theta_0 \sin (\theta_0-\alpha)}{g \cos ^2 \alpha}$

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<footer style = "font-size: 18px">Trajectory Formula $\rightarrow$ Range $\rightarrow$ <span style = "color:blue"> Range </span> $\rightarrow$ Parachute Problems $\rightarrow$ Parachute Problems </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Parachute Problems </primary>
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- $v_m$ = Constant

- $v_A$ = Constant

- Velocity of aircraft = $v_A \hat{i}$

- Missile is always guided towards aircraft

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<footer style = "font-size: 18px">Range $\rightarrow$ Range $\rightarrow$ <span style = "color:blue"> Parachute Problems </span> $\rightarrow$ Parachute Problems $\rightarrow$ Parachute Problems </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Parachute Problems </primary>
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- **Find time taken by missile to hit the aircraft ?**

- Rate of separation of aircraft missile = $\frac{d r}{d t}=\overrightarrow{V_m}-\overrightarrow{V_A}$

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<footer style = "font-size: 18px">Range $\rightarrow$ Parachute Problems $\rightarrow$ <span style = "color:blue"> Parachute Problems </span> $\rightarrow$ Parachute Problems $\rightarrow$ Parachute Problems </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Parachute Problems </primary>
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- $\frac{d(\text { seperation })}{d t}$=$v_m-v_{A, r} $

- $\quad = v_m-v_A \cos \phi$

- $\int d r$=$\int_0^T(v_m-v_A \cos \phi) dt$

- $T \rightarrow$ Time when missile hits the  aircraft

- $\int d r = \sqrt{a_2+b_2}$

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<footer style = "font-size: 18px">Parachute Problems $\rightarrow$ Parachute Problems $\rightarrow$ <span style = "color:blue"> Parachute Problems </span> $\rightarrow$ Parachute Problems $\rightarrow$ Parachute Problems </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Parachute Problems </primary>
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- $\frac{d x}{d t}$=$v_m \cos \phi-v_A $
 
- $\int_0^T d x$=$\int_0^T v_m \cos \phi d t-v_A T $

- a=$v_m \int_0^T \cos \phi d t-v_A T (2) $

-  From (1) we can get $\int_0^T \cos \phi dt$

- substitute in (2)

* T=$\frac{a v_A+v_m \sqrt{a^2+b^2}}{v_m^2-v_{A^2}}$

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Curved Path </primary>
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- $\vec{v}$ is always tangent to the path

* 2 components 
* Tangent to the path $\frac{dv}{dt}$
* $\perp$ path
- points to the centre of curvature

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<footer style = "font-size: 18px">Dot Product $\rightarrow$ Motion $\rightarrow$ <span style = "color:blue"> Curved Path </span> $\rightarrow$ Curved Path $\rightarrow$ Curved Path </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Curved Path </primary>
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- $\vec{v} \cdot \vec{a} $

- $\vec{v} \cdot \frac{d \vec{v}}{d t} $

- =$\frac{d}{d t}$ $(\frac{\vec{v} \cdot \vec{v}} {2}) $
 
- =$\frac{1}{2}$ $\frac{d}{d t} (v^2)$ , $v \rightarrow$   speed
 
 - $\vec{v} .\vec {a}$ =0 
 
 - $\frac {d} {dt} (v^2)$ =0
 
-  speed is not changing with t
 
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![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-04-problemsin-kinematics-motion-in-a-plane-lecture-n96pnciey5y-579-3258.0.jpg)

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<footer style = "font-size: 18px">Motion $\rightarrow$ Curved Path $\rightarrow$ <span style = "color:blue"> Curved Path </span> $\rightarrow$ Curved Path $\rightarrow$ Curved Path </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Curved Path </primary>
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- If for all times

- $\vec{v} \cdot \vec{a}=0$

- $\Rightarrow$ speed is constant

- $\vec{v} \cdot \vec{a}>0$

- $\angle \theta$ between $\vec{v}$ and $\vec{a}$ between 0 and $90^{\circ}$

- $\vec{v} \cdot \vec{a}<0$

- $\angle \theta$ between $90^{\circ}$ and $180^{\circ} $

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<footer style = "font-size: 18px">Curved Path $\rightarrow$ Curved Path $\rightarrow$ <span style = "color:blue"> Curved Path </span> $\rightarrow$ Curved Path $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Problems in Kinematics Motion In A Plane L-4 </Success>
### <primary style="font-size:24px"> Curved Path </primary>
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- $\vec{v} \cdot \vec{a}=\frac{1 d}{2 d t}(v^2) $

- $\vec{v} \cdot \vec{a}>0 \Rightarrow \frac{d}{d t}(v^2)>0$

- Speed is increasing

- $\vec{v} \cdot \vec{a}<0 \Rightarrow$ Speed is decrease

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<footer style = "font-size: 18px">Curved Path $\rightarrow$ Curved Path $\rightarrow$ <span style = "color:blue"> Curved Path </span> $\rightarrow$ Thank You $\rightarrow$  </footer>