Particles in motion..please help

[roxstar1]

A particle, intitially at rest, moves along the x-axis such that its aceleration at time t>0 is given by a(t) = cost. At the time t=0, its position is x=3.
a) find the velocity and position functions for the particle.
b) find the values of t for which the particle is at rest

this is what I have so far...is this right?
a) a(t) = cos (t)
v(t) = sin(t) + c1
-3 = -cos(0) + c2
3 = -1 + c2
4 = c2
s(t) = -cos(t) + 4
v(t) = sin (t)

b) Im lost here...I think I'm supposed to set the position function equal to zero...but Im not sure

any suggestions would be much appreciated

 

[soroban]

Hello, roxstar1!

A particle, intitially at rest, moves along the x-axis
such that its aceleration at time \(\displaystyle t > 0\) is given by \(\displaystyle a(t)\,=\,\cos(t)\).
At the time \(\displaystyle t=0\), its position is \(\displaystyle s=3\)

a) Find the velocity and position functions for the particle.

Your answers are correct, but I can't follow how you got them.

We have: .\(\displaystyle a(t)\:=\:\cos(t)\)
Then: .\(\displaystyle v(t) \:=\:\sin(t)\,+\,c_1\)

The particle begins at rest: .\(\displaystyle v(0)\,=\,0\)
. . So we have: .\(\displaystyle 0\:=\:\sin(0)\,+\,c_1\;\;\Rightarrow\;\;c_1\,=\,0\)
The velocity function is: .\(\displaystyle v(t)\:=\:\sin(t)\)

Then we have: .\(\displaystyle s(t)\:=\:-\cos(t)\,+\,c_2\)
. . Since \(\displaystyle s(0)\,=\,3\), we have: .\(\displaystyle 3\:=\:-\cos(0)\,+\,c_2\;\;\Rightarrow\;\;c_2\,=\,4\)
The position function is: .\(\displaystyle s(t)\:=\:-\cos(t)\,+\,4\)

b) Find the values of \(\displaystyle t\) for which the particle is at rest.
(I think I'm supposed to set the position function equal to zero...but Im not sure)

No, it did <u>not</u> ask "When is the particle at the origin?"

"The particle is at rest" means: \(\displaystyle v(t)\,=\,0\)

 

[roxstar1]

part b

oh, i see, so the particle is at rest when v(t)=0


v(t)=sin (t)

0=sin(t)

t = pi, -pi

is that correct?

 

[Gene]

Half right. You can't go back in time so t=n*pi where n > 0.
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Gene