(a) Find the velocity and position functions of the particle. (b) For what values of time t is the particle at rest?

Question

A particle, initially at rest, moves along the x-axis such that the acceleration at time t>0 is given by a(t)=−sin(t) . At the time t=0 , the position is x=5 .

(a) Find the velocity and position functions of the particle.

(b) For what values of time t is the particle at rest?

Answer 1

a(t) = - sin t

V(t) = cos t + c

initially at rest --> v(0) = 0 ---> cos 0 + c = 0

1+c = 0

c = -1

V(t) = cos t - 1

x(t) = sin t -t + k

x(0) = 5 ---> sin 0 - 0 + k = 5

k = 5

x(t) = sin t - t + 5

V(t) = 0

cos t - 1 = 0

cos t = 1

t = pi