Question

R(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of t.

r(t) = (cos 2t)i + (3sin 2t)j; t = 0

Answer 1

The equation in x and y for the path of the particle is x = cos(2t) and y = 3sin(2t). The velocity vector is v(t) = -2sin(2t)i + 6cos(2t)j. The acceleration vector is a(t) = -4cos(2t)i - 12sin(2t)j.

Step-by-step explanation:

The equation in x and y for the path of the particle can be found by replacing t in the given position vector r(t) = (cos 2t)i + (3sin 2t)j with x and y.

Therefore, the equation in x and y is:

x = cos(2t)

y = 3sin(2t)

To find the velocity vector, we differentiate the position vector with respect to t:

v(t) = -2sin(2t)i + 6cos(2t)j

At t = 0, the velocity vector is:

v(0) = -2sin(0)i + 6cos(0)j = 0i + 6j = 6j

To find the acceleration vector, we differentiate the velocity vector with respect to t:

a(t) = -4cos(2t)i - 12sin(2t)j

At t = 0, the acceleration vector is:

a(0) = -4cos(0)i - 12sin(0)j = -4i