# Frequently used basic physics equations

## 1. Kinematics

These links give the detailed notes on the respective chapters

### Average Velocity and speed

$${v_{avg}} = {{\Delta s} \over {\Delta t}}$$

### Instantaneous velocity and speed

$v = \mathop {\lim }\limits_{\Delta t \to 0} {{\Delta s} \over {\Delta t}} = {{ds} \over {dt}}$
Instantaneous speed or speed is the magnitude of the instantaneous velocity

here $s$ is the displacement of the object and has only one component(out of x, y and z) for motion along straight line and has two components for motion in a plane.

### AccelerationAverage acceleration

$${a_{avg}} = {{\Delta v} \over {\Delta t}}$$
Instantaneous acceleration
$$a = \mathop {\lim }\limits_{\Delta t \to 0} {{\Delta v} \over {\Delta t}} = {{dv} \over {dt}}$$

### Equations of motion (constant acceleration)

$$v = {v_0} + at$$
$$x = {x_0} + {v_0} + {1 \over 2}a{t^2}$$
$${v^2} = {v_0}^2 + 2a(x – {x_0})$$
$$\overline v = {1 \over 2}(v + {v_0})$$

### Free fall acceleration

$$v = {v_0} + gt$$
$$x = {x_0} + {v_0} + {1 \over 2}g{t^2}$$
$${v^2} = {v_0}^2 + 2g(x – {x_0})$$

### Projectiles

Horizontal distance

$$x = {v_x}t$$

Horizontal velocity

$${v_x} = {v_{x0}}$$

Vertical distance

$$y = {v_{yo}}t – {1 \over 2}g{t^2}$$

Vertical velocity

$${v_y} = {v_{y0}} – gt$$

Here,
${v_x}$ is the velocity along x-axis,
${v_{x0}}$ is the initial velocity along x-axis,
${v_y}$ is the velocity along y-axis,
${v_{y0}}$ is the initial velocity along y-axis.
$g$ is the acceleration due to gravity and
$t$ is the time taken.

Time of flight

$$t = {{2{v_o}\sin \theta } \over g}$$

Maximum height reached

$$H = {{v_0^2{{\sin }^2}\theta } \over {2g}}$$

Horizontal range

$$R = {{v_0^2\sin 2\theta } \over g}$$

Here,

$v_{0}$ is the initial Velocity,
${\sin \theta }$ is the component along y-axis,
${\cos \theta }$  is the component along x-axis.

### Uniform circular motion

Angular velocity

$$\omega = {{d\theta } \over {dt}}$$

where $\theta$ is angle moved in radian’s

Relation between linear velocity, angular velocity and radius of circular motion

$$v=r\omega$$

Angular acceleration

$$\alpha = {{d\omega } \over {dt}}$$

Centripetal acceleration

$${a_c} = {{{v^2}} \over r}$$

$${{\vec a}_c} = – {\omega ^2}\vec r$$

## 2. Laws of motion and friction

Links to Full length notes on these chapters are given below

### Newton’s second law of motion

$$\sum F = m\vec a$$

$$\sum F = {{d\vec p} \over {dt}}$$

### Weight

$$W=mg$$

### Limiting friction

$${f_{ms}} = {\mu _s}N$$

where $N$ is the normal contact force and $\mu _s$ coefficient of static friction.

## 3. Work energy and power

Full notes on Work energy and power

### Work

work done by the force F in in displacing the body through displacement d

$$W = \vec F \cdot \vec s$$

$$W = \left| {\vec F} \right|\left| {\vec s} \right|\cos \theta$$

### Work done by variable force

$$W = \int {\vec F \cdot d\vec s}$$

### Kinetic Energy

$$K = {1 \over 2}m{v^2}$$

### Potential Energy

$$\Delta U = – \int {\vec F \cdot d\vec s}$$

$$\vec F = – \nabla U$$

$$F(x) = – {{dU(x)} \over {dx}}$$

### Mechanical energy

$$E = K + U$$

### Gravitational Potential energy

$$U = mgh$$

### Potential energy of the spring

$$F = – kx$$

$$U = {1 \over 2}k{x^2}$$

### Power

$$P = {{dW} \over {dt}}$$

$$\overline P = {{\Delta W} \over {\Delta t}}$$

$$P = \vec F \cdot \vec v$$

### 4. Impulse and Momentum

This link have Impulse and linear momentum notes.

Momentum

$$p = m\vec v$$

Impulse

$$\vec I = \vec F({t_2} – {t_1})$$

$$\vec I = \int\limits_{{t_1}}^{{t_2}} {\vec Fdt}$$

## 5. System of Particles and Collisions

Links to detailed notes on this chapter

### Center of mass position vector

$$\vec R = {{\sum {{m_i}{{\vec r}_i}} } \over M}$$

### Inelastic collision

While colliding if two bodies stick together then speed of the composite body is
$$v = {{{m_1}{u_1} + {m_2}{u_2}} \over {{m_1} + {m_2}}}$$

### Elastic collision in one dimension

Final velocities of bodies after collision are
$${v_1} = \left( {{{{m_1} – {m_2}} \over {{m_1} + {m_2}}}} \right){u_1} + \left( {{{2{m_2}} \over {{m_1} + {m_2}}}} \right){u_2}$$
$${v_2} = \left( {{{2{m_1}} \over {{m_1} + {m_2}}}} \right){u_1} + \left( {{{{m_2} – {m_1}} \over {{m_1} + {m_2}}}} \right){u_2}$$
also
$${u_1} – {u_2} = {v_2} – {v_1}$$

## 6. Rotational Mechanics

Rotational mechanics notes

Angular Velocity

$$\vec v = \vec \omega \times \vec r$$

Angular acceleration

$$\vec a = \vec \alpha \times \vec r – {\omega ^2}\vec r$$

Angular momentum of system of $n$ particles about the origin

$$\vec L = \sum\limits_{i = 1}^n {{{\vec r}_i} \times } {{\vec p}_i}$$

$$L = mvr\sin \theta$$

$$\vec L = I\vec \omega$$

torque or moment of force on system of $n$ particles about the origin

$$\vec \tau = \sum\limits_{i = 1}^n {{{\vec r}_i} \times } {{\vec F}_i}$$

$$\tau = rF\sin \theta$$

Equations of rotation

$$\omega = {\omega _0} + \alpha t$$

$$\theta = {\theta _0} + {\omega _0}t + {1 \over 2}\alpha {t^2}$$

$${\omega ^2} = \omega _0^2 + 2\alpha \left( {\theta – {\theta _0}} \right)$$

$$\bar \omega = {1 \over 2}(\omega + {\omega _0})$$

Moment of inertia

$$I = \sum {m{r^2}}$$

Kinetic energy of rotation

$$K = {1 \over 2}I{\omega ^2}$$

## Gravitation

Universal gravitation

$$F = G{{{m_1}{m_2}} \over {{r^2}}}$$

Orbital speed

$$v = \sqrt {{{Gm} \over r}}$$

escape speed

$$v = \sqrt {{{2Gm} \over r}}$$

Acceleration due to gravity

(1) at height $h$ above the surface of earth

$${g_h} = {{G{m_e}} \over {R_e^2}}\left( {1 – {{2h} \over {{R_e}}}} \right)$$

(2) at depth $d$ below earth’s surface

$${g_d} = {{G{m_e}} \over {R_e^2}}\left( {1 – {d \over {{R_e}}}} \right)$$

Gravitational potential energy

$$V = – G{{{m_1}{m_2}} \over r}$$

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