Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.

검증된 단계별 안내

Recall the definitions: velocity is the rate of change of position with respect to time, and displacement is the change in position over a time interval.

Understand that displacement over a time interval \([a, b]\) can be found by integrating the velocity function \(v(t)\) over that interval: \(\text{displacement} = \int_a^b v(t) \, dt\).

Note that position at a specific time requires an initial position value \(s(a)\), because position is given by \(s(t) = s(a) + \int_a^t v(\tau) \, d\tau\).

Since velocity alone does not provide the initial position \(s(a)\), you cannot determine the exact position function \(s(t)\), only the displacement over intervals.

Therefore, the statement is true: given only velocity, you can find displacement by integration, but you cannot find the exact position without additional information.

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주요 개념

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Velocity and Displacement Relationship

Velocity is the rate of change of position with respect to time, and displacement is the net change in position over a time interval. By integrating velocity over time, one can find the displacement, which measures how far and in what direction the object has moved from its initial position.

Position as an Initial Value Problem

To determine the exact position of an object from its velocity, an initial position value is required. Without knowing where the object started, velocity alone only provides information about changes in position, not the absolute position at any given time.

Definite vs. Indefinite Integration in Kinematics

Integrating velocity to find displacement involves definite integration over a time interval, yielding a specific change in position. Finding position from velocity requires indefinite integration plus an initial condition, highlighting why velocity alone is insufficient to determine position uniquely.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.