For problems that involve an object accelerating along an inclined plane, how can the weight be used to determine the force component that causes motion? a, the weight is the full force that acts perpendicular to the surface of the ramp. b, the weight times the cosine of the ramp’s angle is the force perpendicular to the ramp. c, the weight times the sine of the angle of the ramp acts down along the ramp. d, the weight times the cosine of the angle of the ramp acts down along the ramp

Answer

C, The weight times the sine of the angle of the ramp acts down along the ramp.

Explanation

Assume that the incline is at some angle relative to the ground, we can tell that the weight can be resolved into two components.  The first component of this force is the normal force. The other component is along the incline and it is what is responsible for the objects acceleration. The angle between the weight vector and the normal component is   . If the magnitude of the weigh is , then , by trigonometry, the magnitude of the weight along the incline is .

The correct answer is C, The weight times the sine of the angle of the ramp acts down along the ramp

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.