1.The average velocity of a particle moving in one dimension has a positive value. Is it possible for the instantaneous velocity to have been negative at any time in the interval? Suppose the particle started at the origin x = 0. If its average velocity is positive, could the particle ever have been in the – x region of the axis?

**Ans: **Yes. Yes, if the particle winds up in the +x region at the end.

2. Can the instantaneous velocity of an object at an instantof time ever be greater in magnitude than the average velocity over a time interval containing the instant? Can it ever be less?

**Ans: **Yes. Yes.

3. If an object’s average velocity is nonzero over some time interval, does that mean that its instantaneous velocity is never zero during the interval? Explain your answer.

**Ans:** No. Consider a sprinter running a straight-line race. His average velocity would simply be the length of the race divided by the time it took for him to complete the race. If he stops along the way to tie his shoe, then his instantaneous velocity at that point would be zero.

4. A ball is projected horizontally from the top of a building. One second later, another ball is projected horizontally from the same point with the same velocity. At what point in the motion will the balls be closest to each other? Will the first ball always be travelling faster than the second ball? What will be the time interval between the moments when the two balls hit the ground? Can the horizontal projection velocity of the second ball be changed so that the balls arrive at the ground at the same time?

**Ans:** The balls will be closest together as the second ball is thrown. Yes, the first ball will always be moving faster, since its flight time is larger, and thus the vertical component of the velocity is larger. The time interval will be one second. No, since the vertical component of the motion determines the flight time.

5. The maximum range of a projectile occurs when it is launched at an angle of 45° with the horizontal, if air resistance is ignored. If air resistance is not ignored, will the optimum angle be greater or less than 45.0°? Explain.

**Ans:** The optimal angle would be less than 45°. The longer the projectile is in the air, the more that air resistance will change the components of the velocity. Since the vertical component of the motion determines the flight time, an angle less than 45° would increase range.