Question

five stages - labeled a, b, c, d, and e - of an objects motion are represented by the position-time graph below. during which stage(s) is the object moving away from its starting point? tap on the buttons below to answer. select all that apply.

Explanation:

Step1: Analyze Stage A

In stage A, the position - time graph is horizontal. This means the position of the object does not change with time, so the object is at rest. So it is not moving away from the starting point.

Step2: Analyze Stage B

In stage B, the position of the object is decreasing (moving from positive position towards zero and then negative). So it is moving towards the starting point (since starting point is at position 0, and it is moving from positive to zero and beyond), not away.

Step3: Analyze Stage C

In stage C, the position - time graph is horizontal. The object's position does not change with time, so it is at rest. Not moving away from the starting point.

Step4: Analyze Stage D

In stage D, the object's position is increasing from a negative value towards zero (and then to positive? But in terms of moving away from the starting point (position 0), when moving from negative position towards zero, it is moving towards the starting point? Wait, no. Wait, the starting point is at position 0. If the object is at a negative position (below zero) and moves towards more negative? No, in stage D, the graph is going from the minimum negative position up towards zero? Wait, no, looking at the graph: stage D is from the bottom of C (negative position) up to zero. Wait, no, maybe I misread. Wait, the starting point is at the initial position, which is the positive position in stage A. Wait, the object starts at a positive position (stage A is at positive position, horizontal line). Then in stage B, it moves down through zero (starting point's position) to negative. Then stage C is at negative position (rest), stage D is moving from negative position back towards zero? No, wait, no - the direction of motion: in a position - time graph, the slope gives velocity. But for moving away from the starting point (initial position, which is the positive position in stage A), we need to see when the object's position is getting farther from the initial position (positive) in magnitude. Wait, initial position is at positive (let's say \(x = x_0>0\)). Then:

- Stage A: \(x = x_0\) (constant), so distance from start is 0 (not moving).
- Stage B: \(x\) decreases from \(x_0\) to negative values. So when moving from \(x_0\) to 0, it is moving towards start; when moving from 0 to negative, it is moving away from start (since distance from start (x = 0) is increasing in magnitude, but wait, the start is at \(x = x_0\), not \(x = 0\). Oh! I made a mistake. The starting point is the initial position, which is the position at the start of the motion. Looking at the graph, the initial position (before stage A) is at the positive position (the y - axis intercept of stage A). So stage A is at that initial position (constant, so at rest). Then stage B: moving from initial position (positive) down through \(x = 0\) to negative. So when moving from initial position (positive) to \(x = 0\), it is moving towards the start? No, the start is at the initial position (positive). So moving away from start would be when the position is getting farther from the initial position (positive) in the positive or negative direction? Wait, distance from start is \(|x - x_{start}|\). So:

- Stage A: \(x=x_{start}\), distance = 0 (not moving away).
- Stage B: \(x\) goes from \(x_{start}\) to 0 (distance decreases) then to negative (distance from \(x_{start}\) is \(|x - x_{start}|=x_{start}-x\) when \(x < x_{start}\), and when \(x\) is negative, \(|x - x_{start}|=x_{start}+|x|\), which is increasing as \(|x|\) increases. Wait, maybe a better way: the starting point is at the origin? Wait, the y - axis has 0, and the initial position (stage A) is at positive. Maybe the starting point is at \(x = 0\). Let's re - define: starting point is \(x = 0\). Then:

- Stage A: \(x>0\) (constant), so object is at rest at a position \(x>0\), distance from start (0) is \(x\) (constant, not moving away).
- Stage B: \(x\) decreases from \(x>0\) to \(x < 0\). So when moving from \(x>0\) to 0, it is moving towards start; when moving from 0 to \(x < 0\), it is moving away from start (since distance from 0 is \(|x|\), which is increasing).
- Stage C: \(x\) is constant (negative), so distance from 0 is constant (not moving away).
- Stage D: \(x\) increases from \(x < 0\) to 0. So distance from 0 is decreasing (moving towards start).
- Stage E: \(x = 0\) (constant), at rest, distance from start is 0.

Wait, but maybe the starting point is the initial position (stage A's position). So stage A: at start (rest). Stage B: moving away? No, stage B is moving down. Wait, the problem says "moving away from its starting point". The starting point is the initial position (where it starts, which is the position at the beginning of the motion, i.e., the position at time = 0, which is the position of stage A). So:

- Stage A: position is constant (same as start), so not moving away.
- Stage B: position is decreasing (moving towards a position closer to or beyond the start? Wait, start is at positive, stage B moves to negative, so it is moving away from the start (since it is getting farther from the start position in the negative direction).
- Stage C: position is constant (negative), so not moving away.
- Stage D: position is increasing (moving from negative towards start), so moving towards start.
- Stage E: position is constant (at start? No, stage E is at 0, while start is at positive. Wait, maybe I misinterpret the graph. Let's look at the graph again:

The graph has position on y - axis (positive, 0, negative) and time on x - axis.

- Stage A: horizontal line at positive position (object is at rest at that positive position, which is the starting point? Or is the starting point at 0?

The key is: moving away from starting point means the distance from the starting point is increasing.

If starting point is at 0:

- Stage A: object is at positive position (rest), distance from 0 is positive (constant, not increasing).
- Stage B: object moves from positive to negative. When moving from positive to 0: distance from 0 decreases (moving towards start). When moving from 0 to negative: distance from 0 increases (moving away from start).
- Stage C: object is at negative position (rest), distance from 0 is constant (not increasing).
- Stage D: object moves from negative to 0: distance from 0 decreases (moving towards start).
- Stage E: object is at 0 (rest), distance from 0 is 0 (constant).

If starting point is at the positive position of stage A:

- Stage A: object is at start (rest), distance 0.
- Stage B: object moves to negative position, distance from start (positive) is \(|x - x_{start}|\), which is increasing (since \(x\) is getting more negative, so \(|x - x_{start}|=x_{start}-x\) (since \(x < x_{start}\)) and as \(x\) decreases, \(x_{start}-x\) increases). So stage B: moving away from start.
- Stage C: object is at negative position (rest), distance from start is constant (not moving away).
- Stage D: object moves from negative to 0, distance from start is \(x_{start}-x\), which is decreasing (since \(x\) is increasing towards 0, so \(x_{start}-x\) decreases). So moving towards start.
- Stage E: object is at 0, distance from start is \(x_{start}-0=x_{start}\) (constant, not moving away).

So in either case, stage B is moving away from the starting point? Wait, no, maybe I made a mistake. Wait, the graph: stage B is a line going from the end of stage A (positive position) down to cross 0 and go to negative. So the slope of stage B is negative (velocity is negative). The starting point: if we consider the starting point as the initial position (at the start of time, which is the position of stage A), then when the object moves from stage A's position (positive) to a position that is farther from stage A's position (in the negative direction), it is moving away. So stage B: the object's position is changing from positive (close to start) to negative (far from start), so distance from start is increasing. Stage D: moving from negative (far from start) to 0 (closer to start), so distance from start is decreasing. Stage A: at start (rest). Stage C: at negative (rest, distance from start is constant). Stage E: at 0 (rest, distance from start is constant if start is positive, or 0 if start is 0).

Wait, maybe the starting point is at 0. Then:

- Stage A: object is at \(x = x_1>0\) (rest), distance from 0 is \(x_1\) (constant).
- Stage B: object moves from \(x_1\) to \(x_2<0\). The distance from 0 when at \(x_1\) is \(x_1\), when at \(x_2\) is \(|x_2|\). If \(|x_2|>x_1\), then it is moving away. But we don't know the magnitudes. But the graph shows that stage B is a line going from positive to negative, so the object is moving through 0. So when moving from positive to 0: distance from 0 decreases (moving towards start). When moving from 0 to negative: distance from 0 increases (moving away from start). So part of stage B is moving towards start, part is moving away? But the options are the whole stages.

Wait, maybe the starting point is the origin (0). Then:

- Stage A: object is at positive position (rest) - not moving away.
- Stage B: object moves from positive to negative - so when moving from positive to 0, towards start; from 0 to negative, away from start. But the stage B is a single stage (the line from the end of A to the start of C, crossing 0). So during stage B, the object's position goes from positive to negative, so the distance from 0 first decreases (towards 0) then increases (away from 0). But the question is "during which stage(s) is the object moving away from its starting point". Maybe the starting point is the initial position (stage A's position), so stage A is at start. Then stage B: moving to negative, which is away from start (since start is at positive). Stage D: moving from negative to 0, which is towards start. Stage A: at start. Stage C: at negative (rest). Stage E: at 0 (rest).

Alternatively, maybe the key is that in a position - time graph, moving away from the starting point (origin) means that the position is getting farther from 0 in either positive or negative direction. So:

- Stage A: position is positive (constant) - distance from 0 is constant (not moving away).
- Stage B: position goes from positive to negative - so when moving from positive to 0: distance from 0 decreases (towards start); when moving from 0 to negative: distance from 0 increases (away from start). But the stage is B, so during B, the object is moving through 0, so part of B is towards start, part is away. But the options are the whole stages. Maybe the intended answer is stage B? Wait, no, maybe I messed up. Wait, let's think about velocity and direction. The starting point is 0.

- Stage A: velocity = 0 (rest), position positive.
- Stage B: velocity negative (slope negative), so moving in negative direction. So if starting point is 0, and initial position is positive (stage A), then moving in negative direction: when at positive position, moving towards 0 (towards start); when at negative position, moving away from 0 (start). But the stage B is the entire line from positive to negative, so during stage B, the object moves from positive (close to start) to negative (far from start), so overall, the distance from start (0) first decreases then increases. But the question is "moving away from its starting point" - maybe the starting point is the initial position (stage A's position), so stage A is at start. Then stage B: moving to negative, which is away from start (since start is at positive). Stage D: moving from negative to 0, which is towards start. So the answer is stage B? Wait, but maybe the correct answer is stage B. Wait, no, maybe I made a mistake. Let's check again.

Wait, the graph:

- Stage A: horizontal line (constant position) - object is at rest.
- Stage B: line with negative slope (object moving in negative direction, i.e., towards or away? If starting point is at 0, and stage A is at positive, then moving from positive to negative: first towards 0 (start), then away from 0 (start) when past 0. But the stage is B, so during B, the object is moving through 0, so part of B is towards start, part is away. But the options are the stages, so maybe the intended answer is stage B? Or maybe I misinterpret the starting point.

Alternatively, maybe the starting point is the origin (0), and the object starts at 0. But stage A is at positive, so that can't be. So the object starts at the positive position (stage A's position), which is the starting point. Then:

- Stage A: at start (rest) - not moving away.
- Stage B: moving to negative position - distance from start (positive) is increasing (since it's moving to a position farther from start in the negative direction) - so moving away.
- Stage C: at negative position (rest) - distance from start is constant - not moving away.
- Stage D: moving from negative to 0 - distance from start is decreasing (moving towards start) - not moving away.
- Stage E: at 0 (rest) - distance from start is constant (start is positive, so distance is start's position - 0, which is constant) - not moving away.

So the stage when the object is moving away from its starting point is stage B.

Answer:

B