Question

a particle in 1D motion is subjected to an acceleration a = -alpha . t, where alpha is a positive constant. initial velocity is u at t = 0. the velocity of particle after time t is​

Answer 1

Given:

A particle in 1D motion is subjected to an acceleration $a=-\alpha t$ where $\alpha$ is a positive constant.

To find:

Velocity after time t ?

Calculation:

$\sf \: a = - \alpha t$

$\sf \implies \: \dfrac{dv}{dt} = - \alpha t$

$\sf \implies \: dv = - \alpha \: t \: dt$

$\sf \displaystyle \sf \implies \: \int dv = - \alpha \int t \: dt$

Putting limits:

$\sf \displaystyle \sf \implies \: \int_{0}^{v} dv = - \alpha \int_{0}^{t} t \: dt$

$\boxed{\sf \displaystyle \sf \implies v = - \frac{ \alpha {t}^{2} }{2}}$

Hope It Helps .