An object moves in the xy-plane with coordinates:
x = 4cos(20 t) ( m)
and
y = 4sin(20 t) (m ) .
a/ Find the object’s distance from the the origin and the speed of the object at every instant of time. Deduce
the characteristics of this motion.
b/ Determine the magnitude and the direction of the acceleration vector at each instant of time.
c/ Sketch the trajectory of this object with the velocity vector and acceleration vector at some instant of time t.
Answers
Explanation:
given
x= 4cos(20t)
y= 4sin(20t)
initially at= 0
x= 4cos(20×0)= 4×1= 4 m
y= 4sin(20×0)= 0
therefore the object is at 4 m distance from the origin in the positive direction on x- axis.
speed of the object,
dy/dt= 4× 20 cos(20t)
dx/dt= - 4×20sin(20)t
=> dy/dx= (dy/dt) / ( dx/dt)= -cos(20t)/sin(20t)
Vt= -cot(20t) m/s
acceleration= dVt/dt= 20cosec²(20t) m/s²
An sketch is attached.
Given:
An object moves in the xy-plane with coordinates: x = 4cos(20 t) ( m) and y = 4sin(20 t) (m ) .
To find:
a/ Find the object’s distance from the the origin and the speed of the object at every instant of time. Deduce the characteristics of this motion.
b/ Determine the magnitude and the direction of the acceleration vector at each instant of time.
c/ Sketch the trajectory of this object with the velocity vector and acceleration vector at some instant of time t
Solution:
From given, we have,
x = 4cos(20 t) ( m) and y = 4sin(20 t) (m )
Vx = dx/dt = -80 sin (20t) and Vy = dy/dt = 80 cos (20t)
Ax = dVx/dt = -1600 cos (20t) and Ay = dVy/dt = -1600 sin (20t)
a. The distance from origin,
d = √[x² + y²] = √[16 cos² (20t) + 14 sin² (20t)] = √16 = 4 m
The speed of the object,
Speed = √[Vx² + Vy²] = √[6400 sin² (20t) + 6400 cos² (20t)] = √6400 = 80 m/s
The characteristics of its motion,
The object is moving in a circular path having a radius of 4m moving in counter clockwise direction.
b. The magnitude and the direction of the acceleration
A = √[Ax² + Ay²] = √[(-1600)² cos² (20t) + (-1600)² sin² (20t)] = √1600² = 1600 m/s²
The acceleration is directed towards the origin/centre of the circle.
c. The trajectory of this object with the velocity vector and acceleration vector at some instant of time t is attached below.
