Difference between revisions of "Acceleration"
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| − | Acceleration, the second [[derivative]] of [[displacement]], is defined to be the change of [[velocity]]. | + | ==Definition== |
| + | |||
| + | '''Acceleration''', the second [[derivative]] of [[displacement]], is defined to be the change of [[velocity]] per unit time at a certain instance. | ||
A common misconception is that acceleration implies a POSITIVE change of velocity, while it could also mean a NEGATIVE one. | A common misconception is that acceleration implies a POSITIVE change of velocity, while it could also mean a NEGATIVE one. | ||
| + | ==Formula for Acceleration== | ||
| + | |||
| + | Let <math>\textbf{v}_1</math> be the velocity of an object at a time <math>t_1</math> and <math>\textbf{v}_2</math> be the velocity of the same object at a time <math>t_2</math>. If acceleration, <math>\textbf{a}</math>, is known to be constant, then <cmath>\textbf{a} = \frac{\textbf{v}_2 -\textbf{v}_1 }{t_2 - t_1}</cmath> Note that velocity is a vector, so the magnitudes cannot be just subtracted in general. | ||
| + | |||
| + | If acceleration is not constant, then we can treat velocity as a function of time, <math>v(t)</math>. Then, at a particular instance, <cmath>\textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)</cmath> | ||
| + | |||
| + | ==Useful Formulae== | ||
| + | Position and its time derivatives are often used in kinematics. For example, the following four equations relate the position <math>x</math>, velocity <math>v</math>, and (constant) acceleration <math>a</math> by magnitude: | ||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | x&=x_0+v_0t+\frac{1}{2}at^2 \\ | ||
| + | \Delta x&=\left(\frac{v+v_0}{2}\right)t \\ | ||
| + | v^2&=v_0^2+2a\Delta x \\ | ||
| + | \overline{v}&=\frac{v+v_0}{2}. | ||
| + | \end{align*} | ||
| + | </cmath> | ||
| + | By the chain rule, one can also show | ||
| + | <cmath>a=v\frac{\text{d} v}{\text{d}x}.</cmath> | ||
| + | Lastly, we have the famous formula of Newton relating the force and acceleration experienced by a massive object: | ||
| + | <cmath>\mathbf{F}=m\mathbf{a}.</cmath> | ||
[[Category:Physics]] | [[Category:Physics]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
{{stub}} | {{stub}} | ||
Latest revision as of 13:04, 24 April 2022
Definition
Acceleration, the second derivative of displacement, is defined to be the change of velocity per unit time at a certain instance.
A common misconception is that acceleration implies a POSITIVE change of velocity, while it could also mean a NEGATIVE one.
Formula for Acceleration
Let 
be the velocity of an object at a time 
and 
be the velocity of the same object at a time 
. If acceleration, 
, is known to be constant, then ![\[\textbf{a} = \frac{\textbf{v}_2 -\textbf{v}_1 }{t_2 - t_1}\]](http://latex.artofproblemsolving.com/9/d/0/9d0c5c81651dc2902b5a031eed74255573ad95b1.png)
Note that velocity is a vector, so the magnitudes cannot be just subtracted in general.
If acceleration is not constant, then we can treat velocity as a function of time, 
. Then, at a particular instance, ![\[\textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)\]](http://latex.artofproblemsolving.com/5/e/2/5e2779497134457ee3c1d38bb730e024007acee8.png)
Useful Formulae
Position and its time derivatives are often used in kinematics. For example, the following four equations relate the position 
, velocity 
, and (constant) acceleration 
by magnitude:
By the chain rule, one can also show
![\[a=v\frac{\text{d} v}{\text{d}x}.\]](http://latex.artofproblemsolving.com/0/9/c/09c6558f341007d2d16d301826f9ebcb5d4aacb0.png)
Lastly, we have the famous formula of Newton relating the force and acceleration experienced by a massive object:
![\[\mathbf{F}=m\mathbf{a}.\]](http://latex.artofproblemsolving.com/0/1/3/013a0ec7f2514ce2b0496a8801f7c056a8cafd25.png)
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