Sketch a position-time graph for an object that starts at x = 1.5 m, moves with a velocity of 2.2 m/s from t = 0 to t = 1 s, has a velocity of 0 m/s from t = 1 s to t = 2 s, and has a velocity of –3.7 m/s from t = 2 s to t = 5 s.

The earth remains in orbit around the sun due to the force of gravity. how does the force of gravity exerted by the sun on the earth compare to the force of gravity exerted by the earth on the sun?

As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. there is no one around to , so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. you pull up on the rope to lift the box. use g for the magnitude of the acceleration due to gravity and neglect friction forces. once you have pulled hard enough to start the box moving upward, what is the magnitude f of the upward force you must apply to the rope to start raising the box with constant velocity? express the magnitude of the force in terms of m, the mass of the box.

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k