Answer :
The position is a cosine function, so the particle is at the origin
whenever the cosine is zero.
The first point where the cosine is zero occurs when the angle is π/2 .
That happens when √t = π/2 , so t = π² / 4 is the point where we need
the particle's speed.
Speed is the first derivative of the position.
The derivative with respect to 't' of cos(√t) is [ -1 / (2√t) sin(√t) ] . (chain derivative.)
The speed when [ t = π² / 4 ] is . . .
-1 / 2√(π² / 4) times sin(√(π² / 4)) = -(1 / π) times sin(π/2) = -(1/π) times (1) = -(1/π) .
The first time when the particle is at the origin, it's moving backwards,
into [ -x ] territory, and its speed is (1/π) = about 0.3183... . (rounded)
There you have its speed and direction, so you have its velocity.
whenever the cosine is zero.
The first point where the cosine is zero occurs when the angle is π/2 .
That happens when √t = π/2 , so t = π² / 4 is the point where we need
the particle's speed.
Speed is the first derivative of the position.
The derivative with respect to 't' of cos(√t) is [ -1 / (2√t) sin(√t) ] . (chain derivative.)
The speed when [ t = π² / 4 ] is . . .
-1 / 2√(π² / 4) times sin(√(π² / 4)) = -(1 / π) times sin(π/2) = -(1/π) times (1) = -(1/π) .
The first time when the particle is at the origin, it's moving backwards,
into [ -x ] territory, and its speed is (1/π) = about 0.3183... . (rounded)
There you have its speed and direction, so you have its velocity.
