Question

An object is moving in the x-y plane with the position as a function of time given by →r=x(t)ˆi+y(t)ˆj. Point O is at x=0,y=0 . The object is definitely moving towards O when

A vₓ>0,vᵧ>0
B vₓ<0,vᵧ<0
C xvₓ+yvᵧ<0
D xvₓ+yvᵧ>0

Answer 1

An object is moving toward point O when both its x-velocity and y-velocity are negative.

Step-by-step explanation:

In order for an object to be moving toward point O, both its x-velocity (vₓ) and y-velocity (vᵧ) must be negative. This means that the object's position in both the x and y directions is decreasing over time. Therefore, the correct answer is option B: vₓ<0, vᵧ<0.

Answer 2

The object is moving towards the origin when the scalar product of the position and velocity vectors is negative, answer C xvx + yvy < 0.

Step-by-step explanation:

The object is definitely moving towards the origin Point O when the scalar product of the position vector (→r = x(t)î + y(t)¯) and the velocity vector (V(t) = vx(t)î + vy(t)¯) is negative, so the correct answer is C xvx + yvy < 0. This is because a negative dot product indicates that the vectors are pointing in nearly opposite directions, which would be the case if the object moves toward the origin in the x-y plane.

Considering options A (vx > 0, vy > 0) and B (vx < 0, vy < 0), they suggest motion in a particular direction but do not provide sufficient information about the relationship between the velocity direction and the position with respect to the origin. Option D (xvx + yvy > 0) implies that the velocity vector has a component in the same direction as the position vector, which would indicate moving away from the origin.