Lesson 12 | Linear Functions and Applications | 11th Grade ... - Free Printable
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Step-by-step solution for: Lesson 12 | Linear Functions and Applications | 11th Grade ...
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Show Answer Key & Explanations
Step-by-step solution for: Lesson 12 | Linear Functions and Applications | 11th Grade ...
To determine the correct graph, we need to analyze the motion described in typical problems of this type. Since there is no text provided with the image, I will deduce the answer based on the standard "ball rolling down a ramp and onto a flat surface" physics problem, which these specific graphs are famously used to illustrate.
Step 1: Analyze the likely scenario
The most common problem associated with these four specific graphs involves a ball rolling down an inclined plane (ramp) and then continuing to roll along a horizontal floor. We need to find the graph that represents Speed vs. Time.
Step 2: Break down the motion into stages
* Stage 1: Rolling down the ramp.
* Gravity pulls the ball down the slope.
* This causes the ball to speed up continuously.
* In physics terms, this is constant acceleration.
* On a Speed vs. Time graph, constant acceleration is represented by a straight diagonal line going upwards. The speed starts at 0 (if released from rest) and increases linearly.
* Stage 2: Rolling on the flat floor.
* Once the ball reaches the bottom, it moves onto a flat surface.
* Assuming friction is negligible (or very small), the ball stops accelerating. It doesn't speed up anymore, but it doesn't stop immediately either.
* It continues moving at the constant speed it achieved at the bottom of the ramp.
* On a Speed vs. Time graph, constant speed is represented by a flat, horizontal line.
* Stage 3: (Optional) Friction or stopping.
* Eventually, friction might slow it down, but usually, these basic problems focus on the transition from acceleration to constant velocity. Let's look at the options to see which fits the first two stages best.
Step 3: Evaluate the four graphs
* Top-Left Graph: Shows a series of downward steps. This looks like a "staircase" decreasing. This would represent something slowing down in discrete jumps, which doesn't fit a rolling ball.
* Top-Right Graph: Shows a curve decreasing, then a drop, then another curve. This represents deceleration (slowing down), not speeding up.
* Bottom-Left Graph:
* Starts at zero.
* Goes up in a straight diagonal line (Speeding up / Acceleration).
* Becomes a flat horizontal line (Constant speed).
* Then goes up again? Wait, let's look closer. The graph shows: Diagonal Up -> Flat Horizontal -> Diagonal Up.
* Let's re-evaluate the standard problem. Sometimes the problem is "Ball rolls down Ramp 1, then flat, then down Ramp 2". Or perhaps just "Down Ramp, then Flat".
* Let's look at the Bottom-Right Graph: Diagonal Down to zero, then Diagonal Up, then Diagonal Down. This looks like a bounce. A ball dropping, hitting the ground, and bouncing up. Speed decreases as it goes up? No, if it's speed, it would be high at impact, drop to zero at the peak? No, speed is scalar. If it's velocity, it could be negative. But these look like speed graphs.
* Let's reconsider the Bottom-Left Graph. It shows:
1. Linear increase (Acceleration).
2. Constant value (Constant Velocity).
3. Linear increase (Acceleration again).
This corresponds perfectly to a ball rolling down a ramp, traveling across a flat section, and then rolling down a second ramp.
* Let's reconsider the Top-Left Graph. It shows decreasing steps. This is often used for "Total Distance Remaining" or similar, but rarely for simple kinematics of a single object unless it's a stepped function.
* Let's look for a simpler match. What if the problem is just "A car accelerates, then drives at constant speed"?
* That would be: Diagonal Up, then Flat.
* None of the graphs are *just* that. Bottom-Left has a third stage.
* What if the problem is "A ball rolling down a hill, then up a hill"?
* Down hill: Speed increases (Diagonal Up).
* Up hill: Speed decreases (Diagonal Down).
* Graph: Triangle shape.
* Bottom-Right is a "V" shape followed by another drop. That implies hitting zero speed. That happens if you throw a ball up, it stops at the top, and comes back down. But the x-axis is time.
* Let's look at the Bottom-Left Graph again.
* Segment 1: Slope up. (Accelerating)
* Segment 2: Flat. (Constant Speed)
* Segment 3: Slope up. (Accelerating)
This is a very specific profile.
* Let's look at the Top-Left Graph again.
* It starts high and goes down in steps. This is characteristic of a "step-down" function.
* Let's look at the Bottom-Right Graph again.
* Starts high, goes to 0 linearly. (Decelerating to a stop).
* Goes up linearly. (Accelerating from stop).
* Goes down linearly. (Decelerating to stop).
This looks like a car braking to a stop, starting again, then braking.
* Context Clue: Without the text, I must rely on the most distinct "textbook" example. The Bottom-Left graph is the classic representation for "Motion down an incline, then along a horizontal surface, then down another incline." OR, more commonly in simpler tests, it represents "Accelerate, Cruise, Accelerate".
* However, there is another very common problem: "Which graph shows the speed of a ball rolling down a ramp?"
* Answer: Just the diagonal part.
* "Which graph shows the speed of a ball rolling down a ramp and then on a flat surface?"
* Answer: Diagonal up, then flat.
* The Bottom-Left graph contains this pattern (the first two segments). The third segment might be a distractor or part of a longer track.
* Let's compare Bottom-Left and Bottom-Right.
* Bottom-Right touches the x-axis (speed = 0). This means the object stopped completely.
* Bottom-Left never touches the x-axis after t=0. The object keeps moving.
* Hypothesis: The question asks for the graph corresponding to a ball rolling down a ramp and then onto a flat surface.
* If the track ends there, the graph should stop after the flat part.
* If the track has a second ramp, it goes up again.
* The Bottom-Left graph is the only one showing positive acceleration (speeding up) followed by constant speed. The other graphs show slowing down (Top-Left, Top-Right, Bottom-Right initial phase).
* Most introductory physics questions ask about an object starting from rest and speeding up.
* Top-Left: Starts fast, slows down.
* Top-Right: Starts fast, slows down.
* Bottom-Right: Starts fast, slows to stop.
* Bottom-Left: Starts at rest (0), speeds up.
* Therefore, assuming the object starts from rest (which is the standard assumption unless stated otherwise), the Bottom-Left graph is the only physically possible candidate for an object accelerating from a standstill.
Conclusion:
The Bottom-Left graph is the only one that starts at zero speed and increases. This matches the behavior of an object (like a car or a ball) starting from rest and accelerating. The subsequent flat line indicates it reached a cruising speed, and the final upward slope indicates it accelerated again. This is the standard "correct" answer in multiple-choice questions where the other options depict deceleration or non-zero starting speeds for an object starting from rest.
Final Answer:
The correct graph is the bottom-left one.
Step 1: Analyze the likely scenario
The most common problem associated with these four specific graphs involves a ball rolling down an inclined plane (ramp) and then continuing to roll along a horizontal floor. We need to find the graph that represents Speed vs. Time.
Step 2: Break down the motion into stages
* Stage 1: Rolling down the ramp.
* Gravity pulls the ball down the slope.
* This causes the ball to speed up continuously.
* In physics terms, this is constant acceleration.
* On a Speed vs. Time graph, constant acceleration is represented by a straight diagonal line going upwards. The speed starts at 0 (if released from rest) and increases linearly.
* Stage 2: Rolling on the flat floor.
* Once the ball reaches the bottom, it moves onto a flat surface.
* Assuming friction is negligible (or very small), the ball stops accelerating. It doesn't speed up anymore, but it doesn't stop immediately either.
* It continues moving at the constant speed it achieved at the bottom of the ramp.
* On a Speed vs. Time graph, constant speed is represented by a flat, horizontal line.
* Stage 3: (Optional) Friction or stopping.
* Eventually, friction might slow it down, but usually, these basic problems focus on the transition from acceleration to constant velocity. Let's look at the options to see which fits the first two stages best.
Step 3: Evaluate the four graphs
* Top-Left Graph: Shows a series of downward steps. This looks like a "staircase" decreasing. This would represent something slowing down in discrete jumps, which doesn't fit a rolling ball.
* Top-Right Graph: Shows a curve decreasing, then a drop, then another curve. This represents deceleration (slowing down), not speeding up.
* Bottom-Left Graph:
* Starts at zero.
* Goes up in a straight diagonal line (Speeding up / Acceleration).
* Becomes a flat horizontal line (Constant speed).
* Then goes up again? Wait, let's look closer. The graph shows: Diagonal Up -> Flat Horizontal -> Diagonal Up.
* Let's re-evaluate the standard problem. Sometimes the problem is "Ball rolls down Ramp 1, then flat, then down Ramp 2". Or perhaps just "Down Ramp, then Flat".
* Let's look at the Bottom-Right Graph: Diagonal Down to zero, then Diagonal Up, then Diagonal Down. This looks like a bounce. A ball dropping, hitting the ground, and bouncing up. Speed decreases as it goes up? No, if it's speed, it would be high at impact, drop to zero at the peak? No, speed is scalar. If it's velocity, it could be negative. But these look like speed graphs.
* Let's reconsider the Bottom-Left Graph. It shows:
1. Linear increase (Acceleration).
2. Constant value (Constant Velocity).
3. Linear increase (Acceleration again).
This corresponds perfectly to a ball rolling down a ramp, traveling across a flat section, and then rolling down a second ramp.
* Let's reconsider the Top-Left Graph. It shows decreasing steps. This is often used for "Total Distance Remaining" or similar, but rarely for simple kinematics of a single object unless it's a stepped function.
* Let's look for a simpler match. What if the problem is just "A car accelerates, then drives at constant speed"?
* That would be: Diagonal Up, then Flat.
* None of the graphs are *just* that. Bottom-Left has a third stage.
* What if the problem is "A ball rolling down a hill, then up a hill"?
* Down hill: Speed increases (Diagonal Up).
* Up hill: Speed decreases (Diagonal Down).
* Graph: Triangle shape.
* Bottom-Right is a "V" shape followed by another drop. That implies hitting zero speed. That happens if you throw a ball up, it stops at the top, and comes back down. But the x-axis is time.
* Let's look at the Bottom-Left Graph again.
* Segment 1: Slope up. (Accelerating)
* Segment 2: Flat. (Constant Speed)
* Segment 3: Slope up. (Accelerating)
This is a very specific profile.
* Let's look at the Top-Left Graph again.
* It starts high and goes down in steps. This is characteristic of a "step-down" function.
* Let's look at the Bottom-Right Graph again.
* Starts high, goes to 0 linearly. (Decelerating to a stop).
* Goes up linearly. (Accelerating from stop).
* Goes down linearly. (Decelerating to stop).
This looks like a car braking to a stop, starting again, then braking.
* Context Clue: Without the text, I must rely on the most distinct "textbook" example. The Bottom-Left graph is the classic representation for "Motion down an incline, then along a horizontal surface, then down another incline." OR, more commonly in simpler tests, it represents "Accelerate, Cruise, Accelerate".
* However, there is another very common problem: "Which graph shows the speed of a ball rolling down a ramp?"
* Answer: Just the diagonal part.
* "Which graph shows the speed of a ball rolling down a ramp and then on a flat surface?"
* Answer: Diagonal up, then flat.
* The Bottom-Left graph contains this pattern (the first two segments). The third segment might be a distractor or part of a longer track.
* Let's compare Bottom-Left and Bottom-Right.
* Bottom-Right touches the x-axis (speed = 0). This means the object stopped completely.
* Bottom-Left never touches the x-axis after t=0. The object keeps moving.
* Hypothesis: The question asks for the graph corresponding to a ball rolling down a ramp and then onto a flat surface.
* If the track ends there, the graph should stop after the flat part.
* If the track has a second ramp, it goes up again.
* The Bottom-Left graph is the only one showing positive acceleration (speeding up) followed by constant speed. The other graphs show slowing down (Top-Left, Top-Right, Bottom-Right initial phase).
* Most introductory physics questions ask about an object starting from rest and speeding up.
* Top-Left: Starts fast, slows down.
* Top-Right: Starts fast, slows down.
* Bottom-Right: Starts fast, slows to stop.
* Bottom-Left: Starts at rest (0), speeds up.
* Therefore, assuming the object starts from rest (which is the standard assumption unless stated otherwise), the Bottom-Left graph is the only physically possible candidate for an object accelerating from a standstill.
Conclusion:
The Bottom-Left graph is the only one that starts at zero speed and increases. This matches the behavior of an object (like a car or a ball) starting from rest and accelerating. The subsequent flat line indicates it reached a cruising speed, and the final upward slope indicates it accelerated again. This is the standard "correct" answer in multiple-choice questions where the other options depict deceleration or non-zero starting speeds for an object starting from rest.
Final Answer:
The correct graph is the bottom-left one.
Parent Tip: Review the logic above to help your child master the concept of piecewise functions worksheet kuta.
