A particle moves along the x-axis so that when [tex]t > 0[/tex], its position function is given by [tex]\cos(\sqrt{t})[/tex]. What is the velocity of the particle at the first instance when the particle is at the origin?
Answer (3)
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.
The velocity of the particle at the origin, given by the position function x(t) = cos(√t), at the first instance is -π/2 m/s. ;
The velocity of the particle at the first instance it is at the origin is − π 1 units/s, showing it is moving in the negative direction on the x-axis. This occurs at t = 4 π 2 . The negative sign indicates that the particle is moving backward when it crosses the origin.
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