Acceleration = d/dx (1/2 v^2)

Acceleration = \frac{d}{dx}(\frac{1}{2}v^2):

\displaystyle \boxed{a=\frac{d^2x}{dt^2}=\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2}v^2)},

where x is displacement, v is velocity, and a is acceleration.

Proof:
\begin{aligned}  a&=\frac{dv}{dt} \text{(this is definition of acceleration)}\\  &=\frac{dv}{dx}\cdot\frac{dx}{dt} \text{(chain rule)}\\  &=v\frac{dv}{dx}.  \end{aligned}

Also,
\begin{aligned}  \frac{d}{dx}(\frac{1}{2}v^2)&=v\frac{dv}{dx}. \text{(chain rule)}  \end{aligned}

Therefore, a=\frac{d}{dx}(\frac{1}{2}v^2).

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About mathtuition88

http://mathtuition88.com
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2 Responses to Acceleration = d/dx (1/2 v^2)

  1. What level math is this? Could you demonstrate how the formula works?

    Like

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