Question

A particle moves along the x-axis so that at time t ≥ 0 its position is given by x(t)=cost√t. What is the velocity of the particle at the first instance the particle is at the origin?

Answer 1

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)