What is the solutions of the problems ?

I cant solve this part of the my homework. How we can solve these ?

a)A particle moves along an s axis. Use given information to find the position function of the paticle.
1) a(t)=cost-2t; v(0)=0; s(0)=0
2)a(t)=4cos2t;v(0)=-1;s(0)=-3

1 Answer
Sean
May 3, 2018

Please see below.

Explanation:

.

#color(red)(1))#

#a(t)=cos(-2t); v(0)=0; s(0)=0#

We need to find the velocity function by taking the integral of the acceleration function:

#v(t)=intcos(-2t)dt=-1/2sin(-2t)+C#

Now, we use the given initial conditions to solve for the constant of integration:

#v(0)=-1/2sin(0)+C=0+C=0#

#C=0#

Therefore, our velocity function is:

#v(t)=-1/2sin(-2t)#

We take the integral of the velocity function to find the position function:

#s(t)=int-1/2sin(-2t)dt=-1/2intsin(-2t)dt=-1/2(1/2cos(-2t))+C=-1/4cos(-2t)+C#

Now, we use the given initial conditions to solve for #C#:

#s(0)=-1/4cos(0)+C=-1/4(1)+C=-1/4+C=0#

#C=1/4#

Therefore, our position function is:

#s(t)=-1/4cos(-2t)+1/4#

#color(red)(2))#

#a(t)=4cos2t; v(0)=1; s(0)=3#

#v(t)=int4cos2tdt=4intcos2tdt=4(1/2)sin2t+C=2sin2t+C#

#v(0)=2sin(0)+C=0+C=C=1#

#v(t)=2sin2t+1#

#s(t)=int(2sin2t+1)dt=2intsin2tdt+intdt=2(-1/2)(cos2t)+t+C=-cos2t+t+C#

#s(0)=-cos(0)+0+C=-1+C=3#

#C=4#

#s(t)=-cos2t+t+4#

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