Acceleration

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

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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 Accelerationacceleration centripetal 2

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

 

Tags: Equations for Acceleration