# Acceleration

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

## Acceleration

Acceleration is the rate of change of velocity. Whenever a mass experiences a force, an acceleration is acting.  Acceleration is a vector quantity having magnitude and direction, some of these include displacement, drag, force, lift, momentum, thrust, torque, velocity and weight.

### Acceleration FORMULA

$$\large{ a = \frac { \Delta v } { t } }$$

$$\large{ a = \frac { v_f - v_i } { t } }$$

Where:

$$\large{ a }$$ = acceleration

$$\large{ t }$$ = time

$$\large{ v }$$ = velocity

$$\large{ \Delta v }$$ = velocity differential

$$\large{ v_f }$$ = final velocity

$$\large{ v_i }$$ = initial velocity

Solve for:

$$\large{ v = v_i + at }$$

$$\large{ v_i = v - at }$$

## Acceleration from force

### Acceleration from Force Formula

$$\large{ a = \frac {F}{m} }$$

Where:

$$\large{ a }$$ = acceleration

$$\large{ F }$$ = force

$$\large{ m }$$ = mass

Solve for:

$$\large{ F = ma }$$

$$\large{ m = \frac {F}{a} }$$

## Angular Acceleration

Angular acceleration ( $$\alpha$$ (Greek symbol alpha) ) (also called rotational acceleration) of an object is the rate at which the angle velocity changes with respect to time.

### Angular Acceleration Formula

$$\large{ \alpha = \frac { d \omega } { d t } }$$

$$\large{ \alpha = \frac { \omega_f - \omega_i } { t_f - t_i } }$$

$$\large{ \alpha = \frac { d^2 \theta } { d t^2 } }$$

$$\large{ \alpha = \frac { a_t } { r } }$$

$$\large{ \alpha = \frac { \tau } { I } }$$

Where:

$$\large{ \alpha }$$ (Greek symbol alpha) = angular acceleration

$$\large{ a_t }$$ = lineat tangential path

$$\large{ \theta }$$  (Greek symbol theta) = angular rotation

$$\large{ r }$$ = radius of circular path

$$\large{ t }$$ = time taken

$$\large{ t_f }$$ = final time taken

$$\large{ t_i }$$ = initial time taken

$$\large{ \tau }$$  (Greek symbol tau) = torque

$$\large{ I }$$ = mass moment of inertia or angular mass

$$\large{ \omega }$$  (Greek symbol omega) = angular velocity

$$\large{ \omega _f }$$  (Greek symbol omega) = final angular velocity

$$\large{ \omega _i }$$  (Greek symbol omega) = initial angular velocity

## Average Acceleration

### Average Acceleration FORMULA

$$\large{ \bar {a} = \frac { \Delta v } { \Delta t } }$$

$$\large{ \bar {a} = \frac { v_f - v_i } { t_f - t_i } }$$

Where:

$$\large{ \bar {a} }$$ = average acceleration

$$\large{ \Delta t }$$ = time differential

$$\large{ t_f }$$ = final time

$$\large{ t_i }$$ = initial time

$$\large{ \Delta v }$$ = velocity differential

$$\large{ v_f }$$ = final velocity

$$\large{ v_i }$$ = initial velocity

## Average Angular Acceleration

Average angular acceleration ( $$\bar {\alpha}$$ (Greek symbol alpha) ) of an object is the average rate at which the angle velocity changes with respect to time.

### Average Angular Acceleration Formula

$$\large{ \bar {\alpha} = \frac { \Delta \omega } { \Delta t } }$$

$$\large{ \bar {\alpha} = \frac { \omega_f - \omega_i } { t_f - t_i } }$$

Where:

$$\large{ \bar {\alpha}}$$  (Greek symbol alpha) = average angular acceleration

$$\large{ \Delta t }$$ = time differential

$$\large{ t_f }$$ = final time

$$\large{ t_i }$$ = initial time

$$\large{ \Delta \omega }$$  (Greek symbol omega) = change in  angular velocity

$$\large{ \omega_f }$$  (Greek symbol omega) = final angular velocity

$$\large{ \omega_i }$$  (Greek symbol omega) = initial angular velocity

## Centripetal Acceleration

Centripetal acceleration ( $$a_c$$ ) is acceleration towards the center that keeps an object in an elliptical orbit with the direction of the velocity vector constantly changing.

### Centripetal Acceleration Formula

$$\large{ a_c = \frac { v^2 } { r } }$$

$$\large{ a_c = \frac { \left( r \omega \right) ^2 } { r } }$$

$$\large{ a_c = r \omega^2 }$$

Where:

$$\large{ a_c }$$ = centripetal acceleration

$$\large{ r }$$ = radius

$$\large{ v }$$ = velocity

$$\large{ \omega }$$  (Greek symbol omega) = angular velocity

Solve for:

$$\large{ v = \sqrt { a r } }$$

$$\large{ r = \frac { v^2 } { a } }$$

## Constant Acceleration

Constant acceleration ( $$a_c$$ ) of an object is the constant rate in a straight line at which the velocity changes with respect to time.  These formulas can not be used if acceleration is not constant.

### Constant Acceleration Formula

$$\large{ v_f = v_i \;+\; a_c t }$$

$$\large{ v_f ^2 = v_i ^2 \;+\; 2a_c s }$$

$$\large{ s = \frac { 1 } { 2 } \left( v_f \;+ \; v_i \right) t }$$

$$\large{ s = v_i t \;+\; \frac { 1 } { 2 } a_c t^2 }$$

$$\large{ s = v_f t \;-\; \frac { 1 } { 2 } a_c t^2 }$$

Where:

$$\large{ a_c }$$ = constant acceleration

$$\large{ s }$$ = displacement

$$\large{ t }$$ = time

$$\large{ v_f }$$ = final velocity

$$\large{ v_i }$$ = initial velocity

## Constant Angular Acceleration

Constant angular acceleration ( $$\omega$$ (Greek symbol omega) ) of an object is the constant rate at which the angle velocity changes with respect to time.

### Constant Angular Acceleration Formula

$$\large{ \omega_f = \omega_i \;+\; \alpha t }$$

Where:

$$\large{ \alpha }$$  (Greek symbol alpha) = angular acceleration

$$\large{ t }$$ = time

$$\large{ \omega_f }$$  (Greek symbol omega) = constant angular velocity (final)

$$\large{ \omega_i }$$  (Greek symbol omega) = initial angular velocity

## Gravitational Acceleration

Gravitational acceleration ( $$g$$ ) is the force on an object caused only by gravity.

### Gravitational Acceleration formula

$$\large{ g = \frac {G m} {r^2} }$$

Where:

$$\large{ g }$$ = gravitational acceleration

$$\large{ G }$$ = universal gravitational constant

$$\large{ m }$$ = planet mass

$$\large{ r }$$ = radius from the planet center

Solve for:

$$\large{ m = \frac {g r^2} {G} }$$

$$\large{ r = \sqrt { \frac {G m} {g} } }$$

## Instantaneous Acceleration

Instantaneous acceleration ( $$a_i$$ ) is the acceleration at a particular moment in time along its path.

### Instantaneous Acceleration Formula

$$\large{ a_i = \frac { d v} {d t } }$$

Where:

$$\large{ a_i }$$ = instantaneous acceleration

$$\large{ dt }$$ = time differential

$$\large{ dv }$$ = velocity differential

## Rate of Change in acceleration

The rate of change in acceleration ( $$a_c$$ ) is the change in position or the ratio that shows the relationship of change of an object.

### Rate of Change in acceleration Formula

$$\large{ a_c = \frac {d}{t} }$$

$$\large{ a_c = \frac { a_f \;-\; a_i }{ t } }$$

Where:

$$\large{ a_c }$$ = rate of change in acceleration

$$\large{ a_f }$$ = final acceleration

$$\large{ a_i }$$ = initial acceleration

$$\large{ d }$$ = displacement

$$\large{ t }$$ = time taken for change in velocity

## Tangential Acceleration

Tangential acceleration ( $$a_t$$ ) is how much the tangential velocity of a point at a radius changes with time.

### Tangential Acceleration FORMULA

$$\large{ a_t = r \alpha }$$

$$\large{ a_t = \frac { d \omega } { d t } }$$

Where:

$$\large{ a_t }$$ = tangential acceleration

$$\large{ \alpha }$$  (Greek symbol alpha) = angular acceleration

$$\large{ dt }$$ = time differential

$$\large{ d \omega }$$  (Greek symbol omega) = angular velocity differential

$$\large{ r }$$ = radius of the object rotation