Work Done By Constant Force :

$$\mathbf{W}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{s}}$$

Work Done By Multiple Forces:

$$ \Sigma \vec{F}=\vec{F_1}+\vec{F_2}+\vec{F_3}+\ldots \ldots $$

$$ W=[\Sigma \vec{F}] \cdot \vec{s}$$

$$ W=\vec{F_1} \cdot \vec{s}+\vec{F_2} \cdot \vec{s}+\vec{F_3} \cdot \vec{s}+\ldots$$

$$ W=W_{1}+W_{2}+W_{3}+\ldots$$

Work Done By A Variable Force

The work done $(W)$ by a variable force $(F(x))$ as an object moves from position $x_1$ to position $x_2$ is given by:

$$W = \int_{x_1}^{x_2} F(x) dx$$

Where:

$W$ is the work done. $F(x)$ is the variable force as a function of position x. $x_1$ and $x_2$ are the initial and final positions, respectively.

Relation Between Momentum and Kinetic Energy:

$$\mathrm{K}=\frac{\mathrm{P}^{2}}{2 \mathrm{~m}} \text { and } \mathrm{P}=\sqrt{2 \mathrm{mK}} ; \mathrm{P}=\text { linear momentum }$$

Potential Energy:

$$\int_{U_{1}}^{U_{2}} d U=-\int_{r_{1}}^{r_{2}} \vec{F} \cdot d \vec{r} \quad \text { i.e., } $$

$$U_{2}-U_{1}=-\int_{r_{1}}^{r_{2}} \vec{F} \cdot d \vec{r}=-W$$

$$U=-\int_{\infty}^{r} \vec{F} \cdot d \vec{r}=-W$$

Conservative Forces:

$$\mathrm{F}=-\frac{\partial \mathrm{U}}{\partial \mathrm{r}}$$

Work-Energy Theorem:

$$W_{C}+W_NC+W_PS=\Delta K$$

Modified Form of Work-Energy Theorem:

$$W_{C}=-\Delta U$$

$$W_{NC}+W_{PS}=\Delta K+\Delta U$$

$$W_{NC}+W_{PS}=\Delta E$$

Conservation of Momentum

$$\sum \mathbf{p} _{\text{initial}} = \sum \mathbf{p} _{\text{final}}$$

Power:

The average power ( $\bar{P}$ or $P_{a v}$ ) delivered by an agent is given by $\bar{P}$ or

$$P_{a v}=\frac{W}{t}$$

$$P=\frac{\vec{F} \cdot d \vec{S}}{d t}=\vec{F} \cdot \frac{d \vec{S}}{d t}=\vec{F} \cdot \vec{V}$$

Impulse:

Impulse of a force $F$ action on a body is defined as :-

$$\vec{J}=\int_{t_{i}}^{t_{f}} F d t \quad \vec{J}=\Delta \vec{P}$$

Impulse - Momentum Theorem

Important points :

(i). Gravitational force and spring force are always non-impulsive. (ii). An impulsive force can only be balanced by another impulsive force.

Coefficient Of Restitution (e):

$$e=\frac{\text { Impulse of reformation }}{\text { Impulse of deformation }}=\frac{\int F_{r} d t}{\int F_{d} d t}$$

$$=\frac{\text { Velocity of separation along line of impact }}{\text { Velocity of approach along line of impact }}$$

(a) e=1

$\Rightarrow$ Impulse of Reformation =Impulse of Deformation

$\Rightarrow$ Velocity of separation $=$ Velocity of approach

$\Rightarrow$ Kinetic Energy may be conserved

$\Rightarrow$ Elastic collision.

(b) e = 0

$\Rightarrow$ Impulse of reformation $=0$

$\Rightarrow$ Velocity of separation $=0$

$\Rightarrow$ Kinetic Energy is not conserved

$\Rightarrow$ Perfectly Inelastic collision.

(c) 0 < e <1

$\Rightarrow$ Impulse of Reformation $<$ Impulse of Deformation

$\Rightarrow$ Velocity of separation $<$ Velocity of approach

$\Rightarrow$ Kinetic Energy is not conserved

$\Rightarrow$ Inelastic collision.

Variable Mass System :

If a mass is added or ejected from a system, at rate $\mu \mathrm{kg} / \mathrm{s}$ and relative velocity $\vec{v_{rel}}$ (w.r.t. the system), then the force exerted by this mass on the system has magnitude $\mu\left|\vec{v}_{\text {rel }}\right|$.

Thrust Force $\left(\vec{F}_{t}\right)$:

$$\vec{F_t}=\vec{v_{rel}}\left(\frac{dm}{dt}\right)$$

Rocket Propulsion :

If gravity is ignored and initial velocity of the rocket $u=0$;

$$v=v_{r} \ln \left(\frac{m_{0}}{m}\right)$$