Answer to Question #165332 in Mechanics | Relativity for Opeyemi

Question

A particle moves, so that the distance (s) in meters travel after time (t) seconds is given by s=f(t). Find the expression for velocity and acceleration, if s=cos2πt + sin2πt at t=1 second and π=180°

Answer 1

Displacement of particle as a function of time is given by,

$s(t)=cos2\pi t+sin2\pi t$

Differentiating above equation with respect to time,

$\dfrac{ds}{dt}=-2\pi sin2\pi t+2\pi cos2\pi t$

$\Rightarrow v(t)=2\pi(cos2\pi t-sin2\pi t)$

We get velocity as a function of time.

Differentiating the velocity equation with respect to time,

$\dfrac{dv}{dt}=2\pi(-2\pi sin2\pi t-2\pi sin2\pi t)$

$\Rightarrow a(t)=-4\pi^{2}(sin2\pi t+cos2\pi t)$

We get acceleration as a function of time.

At t = 1 second,

$v(t=1)=2\pi(cos2\pi-sin2\pi)$

$=2\pi(1-0)$

$=2\times3.14$

$v(t=1)=6.28 ms^{-1}$

$a(t=1)=-4\pi^{2}(sin2\pi+cos2\pi)$

$= -4\pi^{2}$

$a(t=1)=-39.47ms^{-2}$