Spring/block on horizontal plane SHM

[physninj]

Homework Statement

1. Find the speed and acceleration when the kinetic energy is equal to half the potential energy
2. Find the minimum displacement between the points where the kinetic energy and acceleration are at half their maximum values

Homework Equations

mv^2=.5kx^2

The Attempt at a Solution

My correct harmonic equations

x(t)=(√2/10)cos(10t+5∏/4)
v(t)=-(√2)sin(10t+5∏/4)
a(t)=-(10*√2)cos(10t+5∏/4)

Got down to for part 1:
arctan(√(1/2))=10t+5∏/4

All I want to know is if its acceptable to use the negative time value that comes from not shifting the output of the inverse tangent, to plug in for velocity and acceleration. Personally I don't see why not.

And for part 2...what the bloody heck are they asking? I suppose I could find the displacements for each of those points and see when they get closest...I don't understand the goal of such a method though.

If you want to see the whole problem I have attached it. Thank you for any help.

 

[haruspex]

for 1, seems fine to me.
For 2, yes, there will be an infinite set of points at which these conditions arise, but there'll be a minimum proximity. Probably something like pi*n*apha+beta and pi*n*alpha+gamma, and you just have to figure out how close such points can get.

 

[physninj]

So is there some way to set up an equation for those displacements, and use a derivative set to zero to find a minimum? Thats the idea I am having right meow anyways

 

[haruspex]

So is there some way to set up an equation for those displacements, and use a derivative set to zero to find a minimum? Thats the idea I am having right meow anyways

No, you won't use differentiation. It's not minimum in a continuous function. It'll be like sin x = .5; x = π/6, 5π/6, 13π/6, ... Smallest difference = 4π/6.