Final answer:
To find the velocity of the particle when it is at the origin, we set x(t) = 0 and solve for t. Then we can use the position function to find the velocity.
Step-by-step explanation:
To find the velocity of the particle, we need to take the derivative of the position function x(t). Using the chain rule, we have:
v(t) = -sin(t)√t - (1/2)cost√t
At the first instance the particle is at the origin (x=0), we set x(t) = 0 and solve for t:
0 = cost√t
cos(t) = 0
t = (2n + 1)(π/2), where n is an integer
Now we can find the velocity at this time:
v(t) = -sin((2n+1)π/2)√((2n+1)π/2) - (1/2)cos((2n+1)π/2)√((2n+1)π/2)