What is the physical quantity represented by the area under this graph? Choose the most appropriate answer. Take note of the quantities represented by the axes.

Question

BlackTom AI · Accepted Answer

Area under a velocity–time graph equals displacement because the integral of velocity with respect to time gives change in position.

1) Relevant knowledge points and concepts
- Velocity v(t) is the rate of change of position x(t): v(t) = dx/dt.
- The definite integral of velocity over a time interval [t1, t2] gives the change in position (displacement): Δx = ∫_{t1}^{t2} v(t) dt.
- Distance is the total path length (integral of speed |v|), position is an absolute location (requires initial x), and change in velocity is given by integrating acceleration, not velocity.

2) Step-by-step reasoning
- The plotted vertical axis is velocity and the horizontal axis is time, so the area under the curve from the initial to final time is ∫ v(t) dt.
- By the fundamental theorem of calculus and the definition of velocity: ∫_{t1}^{t2} v(t) dt = x(t2) − x(t1), which is displacement.

3) Calculation form (intermediate steps)
- Start from v(t) = dx/dt.
- Integrate both sides over [t1, t2]: ∫_{t1}^{t2} v(t) dt = ∫_{t1}^{t2} dx/dt dt.
- Right side simplifies: ∫_{t1}^{t2} dx/dt dt = x(t2) − x(t1) = Δx (displacement).

4) Why the correct answer is correct
- The area directly computes x(t2) − x(t1), the net change in position (displacement). The graph shows positive velocities, so the signed area is positive; thus the displacement is positive.

5) Why other options are incorrect
- (a) Distance: distance is ∫ |v| dt. For strictly positive v the number equals displacement, but “distance” is less general (would differ if v were negative). The question asks the most appropriate physical quantity — displacement is the standard answer for area under v–t.
- (b) Position: area gives change in position, not absolute position (you still need an initial x).
- (d) Change in velocity: that equals ∫ a dt, not ∫ v dt.

6) Conclusion
The area under the velocity–time graph represents displacement. This matches the provided correct answer: Displacement.

What is the physical quantity represented by the area under this graph? Choose the most appropriate answer. Take note of the quantities represented by the axes.单项选择题

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