# 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

$$a = \frac { \Delta v } { t }$$          $$acceleration \;=\; \frac { difference \; in \; velocity } { time }$$

$$a = \frac { v_f - v_i } { t }$$          $$acceleration \;=\; \frac { final \; velocity \; - \; initial \; velocity } { time }$$

Where:

$$a$$ = acceleration

$$t$$ = time

$$v$$ = velocity

$$\Delta v$$ = velocity differential

$$v_f$$ = final velocity

$$v_i$$ = initial velocity

Solve for:

$$v = v_i + at$$

$$v_i = v - at$$

## Acceleration from force

### Acceleration from Force Formula

$$a = \frac {F}{m}$$          $$acceleration \;=\; \frac { force } { mass }$$

Where:

$$a$$ = acceleration

$$F$$ = force

$$m$$ = mass

Solve for:

$$F = ma$$

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

## 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

$$\alpha = \frac { d \omega } { d t }$$          $$angular \; acceleration \;=\; \frac { angular \; velocity } { time \; taken }$$

$$\alpha = \frac { \omega_f - \omega_i } { t_f - t_i }$$           $$angular \; acceleration \;=\; \frac { final \; angular \; velocity \;-\; initial \; angular \; velocity } { final \; time \; taken \;-\; initial \; time \; taken }$$

$$\alpha = \frac { d^2 \theta } { d t^2 }$$          $$angular \; acceleration \;=\; \frac { angular \; velocity } { time \; taken }$$

$$\alpha = \frac { a_t } { r }$$          $$angular \; acceleration \;=\; \frac { lineat \; tangential \; path } { radius \; of \; circular \; path }$$

$$\alpha = \frac { \tau } { I }$$          $$angular \; acceleration \;=\; \frac { torgue } { mass \; moment \; of \; inertia }$$

Where:

$$\alpha$$ (Greek symbol alpha) = angular acceleration

$$a_t$$ = lineat tangential path

$$\theta$$ (Greek symbol theta) = angular rotation

$$r$$ = radius of circular path

$$t$$ = time taken

$$t_f$$ = final time taken

$$t_i$$ = initial time taken

$$\tau$$ (Greek symbol tau) = torque

$$I$$ = mass moment of inertia or angular mass

$$\omega$$ (Greek symbol omega) = angular velocity

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

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

## Average Acceleration

### Average Acceleration FORMULA

$$\bar {a} = \frac { \Delta v } { \Delta t }$$          $$average \; acceleration \;=\; \frac { difference \; in \; velocity } { difference \; in \; time }$$

$$\bar {a} = \frac { v_f - v_i } { t_f - t_i }$$          $$average \; acceleration \;=\; \frac { final \; velocity \;-\; initial \; velocity } { final \; time \;-\; initial \; time }$$

Where:

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

$$\Delta t$$ = time differential

$$t_f$$ = final time

$$t_i$$ = initial time

$$\Delta v$$ = velocity differential

$$v_f$$ = final velocity

$$v_i$$ = 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

$$\bar {\alpha} = \frac { \Delta \omega } { \Delta t }$$          $$average \; angular \; acceleration \;=\; \frac { change \; in \; angular \; velocity } { change \; in \; time }$$

$$\bar {\alpha} = \frac { \omega_f - \omega_i } { t_f - t_i }$$          $$average \; angular \; acceleration \;=\; \frac { final \; angular \; velocity \;-\; initial \; angular \; velocity } { final \;time \;-\; initial \; time }$$

Where:

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

$$\Delta t$$ = time differential

$$t_f$$ = final time

$$t_i$$ = initial time

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

$$\omega_f$$ = final angular velocity

$$\omega_i$$ = 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

$$a_c = \frac { v^2 } { r }$$          $$centripetal \; acceleration \;=\; \frac { velocity \; squared } { radius }$$

$$a_c = \frac { \left( r \omega \right) ^2 } { r }$$          $$centripetal \; acceleration \;=\; \frac { \left( \; radius \;\;x\;\; \omega \; \right) ^2 } { radius }$$

$$a_c = r \omega^2$$          $$centripetal \; acceleration \;=\; radius \;\;x\;\; angular \; velocity^2$$

Where:

$$a_c$$ = centripetal acceleration

$$r$$ = radius

$$v$$ = velocity

$$\omega$$ (Greek symbol omega) = angular velocity

Solve for:

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

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

## 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

$$v_f = v_i \;+\; a_c t$$          $$final \; velocity \;=\; initial \; velocity \;+\; \left(\; constant \; acceleration \;\;x\;\; time \; \right)$$

$$v_f ^2 = v_i ^2 \;+\; 2a_c s$$

$$s = \frac { 1 } { 2 } \left( v_f \;+ \; v_i \right) t$$          $$displacement \;=\; \frac { 1 } { 2 } \left( final; velocity \;+ \; initial \; velocity \right) time$$

$$s = v_i t \;+\; \frac { 1 } { 2 } a_c t^2$$          $$displacement \;=\; initial \; velocity \;\;x\;\; time \;+\; \frac { 1 } { 2 } \;\;x\;\; constant \; acceleration \;\;x\;\; time^2$$

$$s = v_f t \;-\; \frac { 1 } { 2 } a_c t^2$$          $$displacement \;=\; final \; velocity \;\;x\;\; time \;-\; \frac { 1 } { 2 } \;\;x\;\; constant \; acceleration \;\;x\;\; time^2$$

Where:

$$a_c$$ = constant acceleration

$$s$$ = displacement

$$t$$ = time

$$v_f$$ = final velocity

$$v_i$$ = 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

$$\omega_f = \omega_i \;+\; \alpha t$$          $$constant \; angular \; acceleration \;=\; initial \; angular \; velocity \;+\; \left( \; angular \; acceleration \;\;x\;\; time \; \right)$$

Where:

$$\alpha$$ (Greek symbol alpha) = angular acceleration

$$t$$ = time

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

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

## Gravitational Acceleration

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

### Gravitational Acceleration formula

$$g = \frac {G m} {r^2}$$          $$gravitational \; acceleration \;=\; \frac { universal \; gravitational \; constant \;\;x \;\; planet \; mass } { radius \; squared }$$

Where:

$$g$$ = gravitational acceleration

$$m$$ = planet mass

$$r$$ = radius from the planet center

Solve for:

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

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

## Instantaneous Acceleration

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

### Instantaneous Acceleration Formula

$$a_i = \frac { d v} {d t }$$          $$instananeous \; acceleration \;=\; \frac { change \; in \; velocity } { change \; in \; time }$$

Where:

$$a_i$$ = instantaneous acceleration

$$dt$$ = time differential

$$dv$$ = 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

$$a_c = \frac {d}{t}$$          $$rate \; of \; change \; in \; acceleration \;=\; \frac { displacement }{ time }$$

$$a_c = \frac { a_f \;-\; a_i }{ t }$$          $$rate \; of \; change \; in \; acceleration \;=\; \frac { final \; acceleration \;-\; initial \; acceleration } { time \; taken \; for \; change \; in \; velocity }$$

Where:

$$a_c$$ = rate of change in acceleration

$$a_f$$ = final acceleration

$$a_i$$ = initial acceleration

$$d$$ = displacement

$$t$$ = 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

$$a_t = r \alpha$$          $$tangential \; acceleration \;=\; radius \; of \; the \; rotation \;\;x\;\; angular \; acceleration$$

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

Where:

$$a_t$$ = tangential acceleration

$$\alpha$$ (Greek symbol alpha) = angular acceleration

$$dt$$ = time differential

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

$$r$$ = radius of the object rotation