Radius vector!

Classical Mechanics Level 2

A radius vector of a particle varies with time t t as r = a t ( 1 − α t ) r=at\left(1-\alpha t\right) , where a a is a constant vector and α \alpha is a positive factor. Find the acceleration w w of the particle as function of time.

w = a ( 1 − 2 t α ) w=a(1-2t\alpha) w = + 2 a α w=+2a\alpha w = − 2 a α w=-2a\alpha w = 2 t a α w=2ta\alpha

Sahil Silare by Sahil Silare , 20, India

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1 solution

Tapas Mazumdar
Sep 18, 2016

We have,

r → = a t ( 1 − α t ) \overrightarrow{r}=at(1-\alpha t)

Differentiating once w.r.t. time t t , we get,

d r → d t = a ( 1 − 2 α t ) ⟹ v → = a ( 1 − 2 α t ) Here v → represents velocity vector as a function of time t ~~~~~~~~~~\dfrac{d\overrightarrow{r}}{dt}=a(1-2\alpha t) \\ \\ \implies \overrightarrow{v}=a(1-2\alpha t)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small \color{#302B94}{\text{Here}~\overrightarrow{v}~\text{represents velocity vector as a function of time}~t}

Differentiating again w.r.t. time t t , we get,

d v → d t = − 2 a α ⟹ w → = − 2 a α Here w → represents acceleration vector which is independent of time t ~~~~~~~~~~\dfrac{d\overrightarrow{v}}{dt}=-2a\alpha \\ \\ \implies \overrightarrow{w}=\boxed{-2a\alpha} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small \color{#302B94}{\text{Here}~\overrightarrow{w}~\text{represents acceleration vector which is independent of time}~t}

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