I just again need verification on the answers to some questions. Thanks

A particle is moving such that its displacement after t seconds is given by x=3sin2t metres.
a)Find the initial velocity and acceleration
b)Find the maximum displacement
c)Find the times when the particle will be at rest
d)Prove that the acceleration is given by a=-4x

2. Originally Posted by chrisgo
I just again need verification on the answers to some questions. Thanks

A particle is moving such that its displacement after t seconds is given by x=3sin2t metres.
a)Find the initial velocity and acceleration
b)Find the maximum displacement
c)Find the times when the particle will be at rest
d)Prove that the acceleration is given by a=-4x
If you are looking for verification, then I presume you've been able to do at least some work on them? Why don't you tell us what you've been able to do and we can help you better that way.

-Dan

3. i think i am confident with a) and d). But i dont know how to do b) and C). can someone please show me how? and just verify what i have done for a) and d). thanks

a) x=3sin2t
v=dx/dt
v=6cos2t
a=dv/dt
a=-12sin2t

initial velocity: when t=0
v=6cos2t
v=6cos0
v=6m/s
initial acceleration: when t=0
a=-12sin2t
a=-12sin0t
a=0 m/s^-2

d) prove a=-4x
a=-12sin2t, x=3sin2t

-12sin2t=-4(3sin2t)
-12sin2t=-12sin2t
therefore a=-4x

4. Originally Posted by chrisgo
a) x=3sin2t
v=dx/dt
v=6cos2t
a=dv/dt
a=-12sin2t

initial velocity: when t=0
v=6cos2t
v=6cos0
v=6m/s
initial acceleration: when t=0
a=-12sin2t
a=-12sin0t
a=0 m/s^-2
Looks good to me.

Originally Posted by chrisgo
d) prove a=-4x
a=-12sin2t, x=3sin2t

-12sin2t=-4(3sin2t)
-12sin2t=-12sin2t
therefore a=-4x
Okay.

Originally Posted by chrisgo
b)Find the maximum displacement
If you know your trig well (particularly the sine graph), you should be able to answer this without need for calculus.

Otherwise, find a relative maximum of the displacement curve. Since you are dealing with the sine function, the displacement will be the same at every relative maximum, so just pick one. Remember that relative extrema occur only at critical points (i.e., where the derivative is zero or undefined). Find your critical points and apply the first or second derivative test.

Originally Posted by chrisgo
c)Find the times when the particle will be at rest
i.e., find the times when $v = 0$.

5. ## another question

thanks but for the times when the particle will be at rest... You say that V=0. so, 6cos2t=0
Because there is no restricted domain, is there an infinite amount of solutions?

6. Originally Posted by chrisgo
thanks but for the times when the particle will be at rest... You say that V=0. so, 6cos2t=0
Because there is no restricted domain, is there an infinite amount of solutions?
Yes.

$6\cos2t = 0\Rightarrow\cos2t = 0$

$\Rightarrow2t = \frac{(2n + 1)\pi}2$ (i.e., odd multiples of $\pi/2$)

$\Rightarrow t = \frac{(2n + 1)\pi}4,\;n\in\mathbb{Z}$