Question

A particle moves with its position given by x=cos(2t) and y=sin(t), where positions are given in feet from the origin and time t is in seconds.

A)Find the speed of the particle.
Speed = ____________
(include units)

B)Find the first positive time when the particle comes to a stop.
t=_____
(include units)

C)If n is any odd integer, write a formula (in terms of n) for all positive times t at which the particle comes to a stop.
t=______
(include units)


For the first one I got sqrt((-2sin(2t))^2+(cos(t))^2) ft/s.
I don't know ho to calculate the second and third part... please someone help...

Answer 1

x=cos(2t) and y=sin(t)
x'(t)=-2sin(2t),
y'(t)=cos(t)

A.
Speed at time t (in ft/sec)
= sqrt(x'(t)²+y'(t)²)
= sqrt(4sin²(2t)+cos²(t))
= ... simplify as you wish

B.
x'(t)=-2sin(2t)=0
occurs when t=kπ/2, k∈Z

and y'(t)=cos(t)=0
occurs when t=(k+1/2)π, k∈Z

So what is the smallest t when x'(t)=0 AND y'(t)=0?

C.
Work out from B above.

Post if you more hint.