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

\(acceleration \;=\; \frac { difference  \; in \; velocity } { time }   \)

\(a = \frac { \Delta v } { t } \)

\(a = \frac { v_f - v_i } { t } \)

\(a = \frac {F}{m}\)

Where:

\(a\) = acceleration

\(F\) = force

\(m\) = mass

\(t\) = 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

\(acceleration \;=\; \frac { force } { mass }   \)

\(a = \frac {F}{m}\)

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

\(angular \; acceleration \;=\; \frac { angular \; velocity } { time }   \)

\(\alpha = \frac { d \omega } { d t }   \)

\(\alpha = \frac { d^2 \theta } { d t^2 }   \)

\(\alpha = \frac { a_t } { r }   \)

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

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

Average Acceleration

Average Acceleration FORMULA

\(average \; acceleration \;=\; \frac { difference  \; in \;  velocity } { difference \;  in \; time }   \)

\(\bar {a} = \frac { \Delta v } { \Delta t }  \)

\(\bar {a} = \frac { v_f - v_i } { t_f - t_i }   \)

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

\(average \; angular \; acceleration \;=\; \frac { change  \; in \;  angular \; velocity } { change \;  in \; time }   \)

\(\bar {\alpha} = \frac { \Delta \omega } { \Delta t }   \)

\(\bar {\alpha} = \frac {  \omega_f - \omega_i } { t_f - t_i }   \)

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.

Centripetal Acceleration Formula

\(centripetal \; acceleration \;=\; \frac { velocity \; squared } { radius }   \)

\(a_c = \frac { v^2 } { r }   \)

Where:

\(a_c\) = centripetal acceleration

\(r\) = radius

\(v\) = 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

\( final \; velocity \;=\; initial \; velocity \;+\;  \left(\; constant \; acceleration \;\;x\;\;  time \; \right)   \)

\( v_f = v_i \;+\; a_c t   \)

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

\(s =  v_i t \;+\;  \frac { 1 } { 2 } a_c t^2  \)

\(s =  v_f t \;-\;  \frac { 1 } { 2 } a_c t^2  \)

\(v_f ^2 =  v_i ^2  \;+\;  2a_c s   \)

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

\(constant \; angular \; acceleration \;=\; initial \; angular \; velocity \;+\; \left( \; angular \; acceleration  \;\;x\;\;  time \; \right) \)

\(\omega_f =  \omega_i  \;+\; \alpha t  \)

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

\(gravitational \; acceleration \;=\; \frac { universal \; gravitational \; constant  \;\;x \;\; planet \; mass } { radius \; squared }   \)

\(g = \frac {G m} {r^2} \)

Where:

\(g\) = gravitational acceleration

\(G\) = universal gravitational constant

\(m\) = 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

\(instananeous \; acceleration \;=\;  \frac { change \; in \; velocity } { change \; in \; time }  \)

\(a_i = \frac { d v} {d t }   \)

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

\(rate \; of \; change \; in \; acceleration \;=\; \frac { displacement }{ time }\)

\(a_c = \frac {d}{t}\)

\(a_c = \frac { a_f \;-\; a_i }{ t }\)

Where:

\(a_c\) = rate of change in acceleration

\(a_f\) = final acceleration

\(a_i\) = initial acceleration

\(d\) = displacement

\(t\) = time taken tor 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

\(tangential \; acceleration \;=\; radius \; of \; the \; rotation \;\;x\;\; angular \; acceleration  \)

\(a_t = r \alpha \)

\(a_t = \frac { d \omega } { d t } \)

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

Tangential Speed FORMULA

\(tangential \; speed \;=\; radius \; of \; the \; rotation \;\;x\;\; angular \; velocity  \)

\(v_t = r \omega  \)

Where:

\(r\) = radius of the object rotation

\(v_t\) = tangential speed

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

 

Tags: Equations for Acceleration