Question

An object moves in the xy-plane with coordinates:
x= 4cos(20t) (m) and y=4sin(20t) (m).
a/ Deduce the characteristics of this motion.
b/ Sketch the trajectory of this object with the velocity vector and acceleration vector at some instant of time t.

Answer 1

The values of acceleration vector is ax = - 1600 cost 20 t and ay = - 160 sin 20 t

Explanation:

The characteristic of this motion is circular motion.

For speed in x-direction:

Vx = dx / dt = d ( 4cos 20 t ) / dt

Vx = - 20 x 4 sin 20 t

Vx = - 80 sin (20 t)

For speed in y-direction:

Vy = dy / dt = d ( 4 sin 20 t) / dt

Vy = 20 x 4 cos 20 t = 80 cos (20 t)

The speed is:

V = √ Vx^2 + Vy^2

V = √ (- 80 sin (20 t)^2 + (80 cos 20 t)^2

V = 80

Acceleration in x-direction:

ax = dVx / dt = d ( -80 sin 20 t) / dt

ax = - 80 x 20 cos 20 t

ax = - 1600 cost 20 t

Acceleration in y-direction:

ay = dVy / dt = d ( 80 cos 20 t) / dt

ay = - 20 x 80 sin 20 t

ay = - 160 sin 20 t