Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

검증된 단계별 안내

Recall the definitions: displacement on an interval is the net change in position, calculated as \(s(b) - s(a)\) where \(s(t)\) is the position function, and distance traveled is the total length of the path traveled, which accounts for all movement regardless of direction.

Understand that velocity being positive on an interval means the object is moving forward (increasing position) throughout that interval, so \(v(t) > 0\) for all \(t\) in the interval.

Since velocity is positive, the position function \(s(t)\) is strictly increasing on that interval, meaning the object never moves backward.

Because the object only moves forward, the displacement (net change in position) equals the total distance traveled, as there are no backward movements to add extra distance.

Therefore, when velocity is positive on an interval, displacement and distance traveled on that interval are indeed equal.

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주요 개념

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Velocity and Its Significance

Velocity is the rate of change of position with respect to time and includes direction, indicated by its sign. A positive velocity means the object is moving in the positive direction, which affects how displacement and distance are calculated over an interval.

Displacement vs. Distance Traveled

Displacement is the net change in position from the start to the end of an interval, considering direction, while distance traveled is the total length of the path covered, always non-negative. They are equal only if the motion does not change direction.

Relationship Between Velocity and Displacement/Distance

If velocity remains positive over an interval, the object moves continuously in one direction, so displacement equals distance traveled. However, if velocity changes sign, displacement and distance differ, highlighting the importance of velocity's sign in comparing these quantities.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.