physics-class-11-unit-06-chapter-05-impact-and-collision-lecture-5-6-jz0-yev-lu0

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### <primary style="font-size:24px"> Impact and Collision </primary>
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- We will consider problems of <span style="color:green">impact and collision</span>.

- Body one is moving with speed $v_1$, and body two is moving with a speed of $v_2$.

- They touch each other: <span style="color:blue">Impact</span>

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<footer style = "font-size: 18px"> $\rightarrow$ Impact and Collision $\rightarrow$ <span style = "color:blue"> Impact and Collision </span> $\rightarrow$ Impact and Collision $\rightarrow$ Impact </footer>

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- Two bodies, one and two of mass $m_1$ and $m_2$ impact each other and move after the impact.

- <span style="color:green">Impact - COLLISION</span>

- The bodies are approaching each other, can we find the final velocities of these bodies using the <span style="color:green">laws of mechanics</span>.

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<footer style = "font-size: 18px">Impact and Collision $\rightarrow$ Impact and Collision $\rightarrow$ <span style = "color:blue"> Impact and Collision </span> $\rightarrow$ Impact $\rightarrow$ Impact </footer>

### <Success style="font-size:25px"> Impact And Collision L-5 </Success>
### <primary style="font-size:24px"> Line of Impact </primary>
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- $t \rightarrow$ <span style="color:blue">tangent plane of the bodies</span>.

- In two dimension, $t$ will be a line.

- Perpendicular to $t$ in the plane.

- $n$ direction is the <span style="color:blue">line of impact</span>.

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<footer style = "font-size: 18px">Impact $\rightarrow$ Impact $\rightarrow$ <span style = "color:blue"> Line of Impact </span> $\rightarrow$ Impact $\rightarrow$ Collision Force </footer>

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- When we have two impacting bodies, we identify

- <span style="color:green">t</span> $\rightarrow$ <span style="color:green">as the plane of contact</span>.

- <span style="color:blue">n</span> $\rightarrow$ <span style="color:blue">line of impact</span>.

- **Smooth impact** $\Rightarrow$ impact force on each body separately is along the line of impact.

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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-06-chapter-05-impact-and-collision-lecture-5_6-jz0-yev-lu0-115-0502.8.jpg)

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<footer style = "font-size: 18px">Impact $\rightarrow$ Line of Impact $\rightarrow$ <span style = "color:blue"> Impact </span> $\rightarrow$ Collision Force $\rightarrow$ Direct or Head-on Collision </footer>

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### <primary style="font-size:24px"> Direct or Head-on Collision </primary>
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- If $\overrightarrow{v_{1 i}}$ and $\vec{v}_{2 i}$ are only along $n$ direction, then the collision is **head-on / direct collision.**

- How to decide if a collision is head-on or a direct collision?

- Draw the tangent plane, $t \rightarrow (n \perp t)$.

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<footer style = "font-size: 18px">Collision Force $\rightarrow$ Direct or Head-on Collision $\rightarrow$ <span style = "color:blue"> Direct or Head-on Collision </span> $\rightarrow$ Direct or Head-on Collision $\rightarrow$ Direct Collision/Head-on Collision </footer>

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### <primary style="font-size:24px"> Analysis </primary>
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- <span style="color:green">Impulse on body 1</span> = $\int F dt$.

- <span style="color:green">Impulse on body 2</span> = -$\int F dt$.

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<footer style = "font-size: 18px">Direct or Head-on Collision $\rightarrow$ Direct Collision/Head-on Collision $\rightarrow$ <span style = "color:blue"> Analysis </span> $\rightarrow$ Direct Collision/Head-on Collision. $\rightarrow$ Direct Collision/Head-on Collision </footer>

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- If we apply Impulse momentum principle on body 1:

- $\overrightarrow{v_{1 i}}$ is only along $n$.

- $ m_1 \overrightarrow{v_{1 i}}+\overrightarrow{I} {\=} m_1 \overrightarrow{v_{1 f}}$

- <span style="color:green">Initially momentum:</span> $m_1 \overrightarrow{v_{1 i}}$

- <span style="color:green">Finally momentum:</span> $m_1 \overrightarrow{v_{1 f}}$

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<footer style = "font-size: 18px">Direct Collision/Head-on Collision $\rightarrow$ Analysis $\rightarrow$ <span style = "color:blue"> Direct Collision/Head-on Collision. </span> $\rightarrow$ Direct Collision/Head-on Collision $\rightarrow$ Oblique Collision </footer>

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* If $\overrightarrow{v_{1 i}}$ only along $n$ i.e., head on collision,

- <span style="color:green"> $t$ component of $\vec{v}_{l i}=0$</span>

* Impulse is also along $\hat{n}$,$\overrightarrow{v_{1 f}}$ is also along $\hat{n}$

- <span style="color:green"> $t$ component of $\overrightarrow{v}_{1 f}=0$.</span>

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<footer style = "font-size: 18px">Analysis $\rightarrow$ Direct Collision/Head-on Collision. $\rightarrow$ <span style = "color:blue"> Direct Collision/Head-on Collision </span> $\rightarrow$ Oblique Collision $\rightarrow$ Oblique Collision </footer>

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- Both the bodies will move with the same velocity just after collision.

- $\vec{v}_{1 i} \rightarrow $ <span style="color:green">n component and t component</span> $\neq 0$.

- Impulsive force is only along n.

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<footer style = "font-size: 18px">Direct Collision/Head-on Collision. $\rightarrow$ Direct Collision/Head-on Collision $\rightarrow$ <span style = "color:blue"> Oblique Collision </span> $\rightarrow$ Oblique Collision $\rightarrow$ Direct Collision </footer>

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- $t$ component of momentum of body 1 will not change, because impulse on the body 1 is only along $n$

- $(v_ {1t}) _ i = (v_ {1t})_f$

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<footer style = "font-size: 18px">Direct Collision/Head-on Collision $\rightarrow$ Oblique Collision $\rightarrow$ <span style = "color:blue"> Oblique Collision </span> $\rightarrow$ Direct Collision $\rightarrow$ Direct Collision </footer>

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- Same arguments hold for body 2.

- <span style="color:green">$\vec{v{ }_{2 i}}$ (intial velocity)</span>

- <span style="color:green">$\vec{v}{ }_{2 f}$ (final velocity)</span>

- Since Impulse is only along $\hat{n}$ :

- <span style="color:blue">$(v_ {2t}) _ i = (v_ {2t})_f$</span>

- Our unknowns are $v_{1 f}$ and $v_{2 f}$(which are along n).

- $-m_2 v_2 i+{I} =m_2 v_{2 f}$

- **Alternate:** Momentum equation in the $n$ direction for both the bodies together.

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<footer style = "font-size: 18px">Oblique Collision $\rightarrow$ Oblique Collision $\rightarrow$ <span style = "color:blue"> Direct Collision </span> $\rightarrow$ Direct Collision $\rightarrow$ Direct Collision </footer>

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- Initial momentum along $n$ direction for both the bodies: $m_1 v_{1 i}-m_2 v_{2 i}$

- $v_{1 f}$  and $v_{2 f}$<span style="color:green">(positive along n).</span>

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<footer style = "font-size: 18px">Oblique Collision $\rightarrow$ Direct Collision $\rightarrow$ <span style = "color:blue"> Direct Collision </span> $\rightarrow$ Direct Collision $\rightarrow$ Coefficient of Restitution </footer>

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### <primary style="font-size:24px"> Elastic collision </primary>
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- Elastic collision: $ e=1$

- Kinetic energy of the two bodies together as system is conserved before and after collision.

- If a collision is not elastic then <span style="color:blue">(kinetic energy post collision) $<$ (kinetic energy pre collision)</span>.

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<footer style = "font-size: 18px">Direct Collision $\rightarrow$ Coefficient of Restitution $\rightarrow$ <span style = "color:blue"> Elastic collision </span> $\rightarrow$ Non- Elastic Collisions $\rightarrow$ Coefficient of Restitution </footer>

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- Lost energy in non elastic collisions may be converted to sound / internal energy of the two bodies show up as heat.

- Completely inelastic collision $e=0$

- $v_{\text {1 f}}$ and $v_{2 f}$ in n direction are equal.

- n component of $v_{1 f}$ = n component of $v_{2_f}$

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<footer style = "font-size: 18px">Coefficient of Restitution $\rightarrow$ Elastic collision $\rightarrow$ <span style = "color:blue"> Non- Elastic Collisions </span> $\rightarrow$ Coefficient of Restitution $\rightarrow$ Separation and approach </footer>

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- e = Coefficient of Restitution

- **$e=\frac{- \text { (elative velocity of separation) }}{\text { (relative velocity of approach) }}$**

- Separation is always post impact and approach is pre-impact.

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<footer style = "font-size: 18px">Elastic collision $\rightarrow$ Non- Elastic Collisions $\rightarrow$ <span style = "color:blue"> Coefficient of Restitution </span> $\rightarrow$ Separation and approach $\rightarrow$ Relative Velocity of Separation </footer>

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- **For given problem**

- <span style="color:blue">Relative velocity of separation</span>$=v_{\text {i f}}-v_{2 f}$

- <span style="color:blue">Relative velocity of approach</span> $=v_{1 i}-\left(-v_{2 i}\right)$

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<footer style = "font-size: 18px">Separation and approach $\rightarrow$ Relative Velocity of Separation $\rightarrow$ <span style = "color:blue"> Separation and approach </span> $\rightarrow$ Separation and approach $\rightarrow$ Separation and approach </footer>

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- **$e=\frac{- \text { (relative velocity of separation) }}{\text { (relative velocity of approach) }}$**

* $e=\frac{-(v_{1 f}-v_{2 f})}{(v_{1 i}+v_{2 i})}$

* $e\left(v_{1 i}+v_{2 i}\right)=v_{2 f}-v_{1 f}$

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<footer style = "font-size: 18px">Relative Velocity of Separation $\rightarrow$ Separation and approach $\rightarrow$ <span style = "color:blue"> Separation and approach </span> $\rightarrow$ Separation and approach $\rightarrow$ Separation and approach </footer>

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- $m_1 v_{1 i}-m_2 v_{2 i}=m_1 v_{1 f}+m_2 v_{2 f}$ ...(1)

- $e=\frac{v_{2 f}-v_{1 f}}{v_{2 i}+v_{1 i}}$

- $v_{2 f}=v_{1 f}+e(v_{2 i}+v_{1 i})$ ...(2)

- **Substitution in (1)**

- $m_1 v_{1 i}-m_2 v_{2 i}=m_1 v_{1 f}+m_2(v_{1 f}+e v_{2 i}+e v_{1 i})$

- $e\left(v_{1 i}+v_{2 i}\right)=v_{2 f}-v_{1 f}$

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<footer style = "font-size: 18px">Separation and approach $\rightarrow$ Separation and approach $\rightarrow$ <span style = "color:blue"> Separation and approach </span> $\rightarrow$ Separation and approach $\rightarrow$ Collision </footer>

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- $m_1=m_2$

- Elastic collision $e=1$

- Platic collision or totally inelastic collision $e=0$

- $m_1 v_{1 i}-m_2 v_{2 i}=\left(m_1+m_2\right) v_{2 f}$

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<footer style = "font-size: 18px">Separation and approach $\rightarrow$ Separation and approach $\rightarrow$ <span style = "color:blue"> Collision </span> $\rightarrow$ Collision $\rightarrow$ Collision </footer>

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- If we write the equation of conservation of energy:

- **$\frac{1}{2} m_1 v_{1 i}^2=\frac{1}{2} m_1 v_{1 f}^2+\frac{1}{2} m_2 v_{2 f}^2$**

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<footer style = "font-size: 18px">Collision $\rightarrow$ Oblique Elastic Collision $\rightarrow$ <span style = "color:blue"> Oblique Elastic Collision </span> $\rightarrow$ Oblique Elastic Collision $\rightarrow$ Oblique Elastic Collision </footer>

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- **$v_{1 i}^2=v_{1 f}^2+v_{2 f}^2$**

- Three form right angle triangle.

- $v_{i f}$ has to be perpendicular to $v_{2 f}$.

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<footer style = "font-size: 18px">Oblique Elastic Collision $\rightarrow$ Oblique Elastic Collision $\rightarrow$ <span style = "color:blue"> Oblique Elastic Collision </span> $\rightarrow$ Oblique Elastic Collision $\rightarrow$ Oblique Elastic Collision </footer>

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- Body $m_2$ is very massive

- <span style="color:green">$m_2 \gg m_1$</span>

- $m_2$ could be earth.

- $v_{1 f}=\frac{m_1 v_{1 i}-m_2\left(v_{2 i}+e v_{2 i}+e v_{1 i}\right)}{\left(m_1+m_2\right)}$

- Divide RHS  (Numerator & Denominator  by $\mathrm{m}_2$ )

- $v_{i f}=\frac{\frac{m_1}{m_2} v_{1 i}-\left(v_{2 i}+e v_{2 i}+e v_{1 i}\right)}{\left(\frac{m 1}{m_2}+1\right)  If v_{2 i}=0}$.

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- $v_{2 i}=0$

- $v_{1 f}=-e v_{1 i}$

- $v_{2 f}=-v_{2 i}$

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