The position vector describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point .

Question

The position vector  describes the path of an object moving in the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point  .

Quizplus · Accepted Answer

Since the position vector is in the xy-plane, the path of the object can be represented as a curve in two dimensions.
At the given point, the velocity vector will be tangent to the curve and the acceleration vector will be tangent to the velocity vector and will point towards the center of curvature of the curve.
The given position vector in coordinates (x,y) is (6374cos(40t), 6374sin(40t)). Therefore, the curve traced out by the object is a circle with radius 6374. Since the object is moving counterclockwise, the velocity vector will point tangent to the circle in the counterclockwise direction. Based on the properties of circular motion, the acceleration vector will point towards the center of the circle, which can be found by taking the cross product of the unit tangent vector to the curve with the unit normal vector (in the counterclockwise direction) to the curve.
At the point given, the unit tangent vector will be pointing vertically downward and the unit normal vector will be pointing horizontally to the right. Therefore, the acceleration vector will point towards the center of the circle in the upward direction.
The correct choice is C, 11eaa595_6385_e203_a696_2f7aee220a56_TB4584_11_TB4584_11, which corresponds to an upward pointing vector in the xy-plane.

The Position Vector Describes the Path of an Object