Difference between revisions of "Acceleration"

(Formula for Acceleration)
 
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Acceleration, the second [[derivative]] of [[displacement]], is defined to be the change of [[velocity]].
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==Definition==
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'''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.
  
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==Formula for Acceleration==
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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.
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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>
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==Useful Formulae==
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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:
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<cmath>
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\begin{align*}
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x&=x_0+v_0t+\frac{1}{2}at^2 \\
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\Delta x&=\left(\frac{v+v_0}{2}\right)t \\
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v^2&=v_0^2+2a\Delta x \\
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\overline{v}&=\frac{v+v_0}{2}.
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\end{align*}
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</cmath>
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By the chain rule, one can also show
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<cmath>a=v\frac{\text{d} v}{\text{d}x}.</cmath>
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Lastly, we have the famous formula of Newton relating the force and acceleration experienced by a massive object:
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<cmath>\mathbf{F}=m\mathbf{a}.</cmath>
 
[[Category:Physics]]
 
[[Category:Physics]]
 
[[Category:Definition]]
 
[[Category:Definition]]
  
 
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Latest revision as of 14:04, 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 $\textbf{v}_1$ be the velocity of an object at a time $t_1$ and $\textbf{v}_2$ be the velocity of the same object at a time $t_2$. If acceleration, $\textbf{a}$, is known to be constant, then \[\textbf{a} = \frac{\textbf{v}_2 -\textbf{v}_1 }{t_2 - t_1}\] 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, $v(t)$. Then, at a particular instance, \[\textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)\]

Useful Formulae

Position and its time derivatives are often used in kinematics. For example, the following four equations relate the position $x$, velocity $v$, and (constant) acceleration $a$ by magnitude: \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*} By the chain rule, one can also show \[a=v\frac{\text{d} v}{\text{d}x}.\] Lastly, we have the famous formula of Newton relating the force and acceleration experienced by a massive object: \[\mathbf{F}=m\mathbf{a}.\]

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