TX S.6.8.A NGSS strat

Students in Ms. Rivera's class used a spring launcher to push a 0.8 kg car along a low-friction track at five different speeds. A photogate sensor measured the car's speed at the end of the track, and students calculated the kinetic energy.

The scatter plot curves upward instead of forming a straight line. Using at least two data intervals from the graph, explain why the KE increases by larger amounts at higher speeds.

Answer explanation

Claim: The kinetic energy of the car increases at a faster and faster rate as speed increases, producing an upward curve rather than a straight line. Evidence: From 1.5 to 2.5 m/s, KE increases by 1.6 J (0.9 to 2.5 J). From 3.5 to 4.5 m/s, KE increases by 3.0 J (5.0 to 8.0 J). From 4.5 to 5.5 m/s, KE increases by 4.2 J (8.0 to 12.2 J). Each successive 1 m/s interval produces a larger jump in KE. Reasoning: This pattern occurs because kinetic energy depends on the square of speed (v²), not on speed directly. When speed goes from 1.5 to 2.5 m/s, v² changes from 2.25 to 6.25 — a change of 4.0. When speed goes from 4.5 to 5.5 m/s, v² changes from 20.25 to 30.25 — a change of 10.0. Since the change in v² grows at higher speeds, the change in KE also grows, producing the upward curve rather than a straight line.

Tags

S.6.8.A

DOK 2

Blooms: Apply

Relationship of KE to speed

Students in Ms. Okafor's class adjusted the fan speed on a 0.5 kg fan cart running on a level air track. A motion sensor recorded the cart's top speed at each fan setting, and students calculated the kinetic energy.

A student predicts that at 8.0 m/s the fan cart would have about 13 J of kinetic energy, reasoning that 'each additional m/s adds roughly 2 J.' Using the graph's pattern, explain whether this prediction is too high, too low, or correct.

Answer explanation

Claim: The prediction of 13 J is too low because it assumes a constant (linear) increase per m/s, while the graph shows KE increases by larger and larger amounts at higher speeds. Evidence: The graph shows the increments are not constant: from 2.0 to 3.0 m/s, KE increases by 1.3 J; from 4.0 to 5.0 m/s, it increases by 2.3 J; from 5.0 to 6.0 m/s, it increases by 2.7 J. Each successive interval adds more KE than the last. Reasoning: Because kinetic energy depends on the square of speed, the increments continue to grow at higher speeds. Beyond 6.0 m/s, each additional m/s would add well more than 2.7 J. Using a constant 2 J per m/s underestimates the growth significantly. The actual KE at 8.0 m/s would be closer to 16 J (since the v² pattern predicts ½ × 0.5 × 64 = 16 J), confirming that 13 J is too low.

Tags

S.6.8.A

DOK 2

Relationship of KE to mass and speed

Blooms: Analyze

Students in Ms. Yamada's class stretched a rubber band to five different lengths to launch a 0.3 kg marble across a smooth hallway floor. A motion sensor measured the marble's speed after each launch, and students calculated the kinetic energy.

Damien says the marble at 10.0 m/s should have 25 times the KE of the marble at 2.0 m/s because 'speed is 5 times greater and KE goes as speed squared, so 5 squared equals 25.' Check his claim using the graph data and explain whether it holds.

Answer explanation

Claim: Damien's reasoning is correct — the marble at 10.0 m/s has approximately 25 times the KE of the marble at 2.0 m/s, consistent with the v² relationship. Evidence: From the graph, KE at 2.0 m/s is 0.6 J and KE at 10.0 m/s is 15.2 J. The ratio is 15.2 ÷ 0.6 ≈ 25.3. Reasoning: Because kinetic energy depends on the square of speed, multiplying speed by 5 (from 2.0 to 10.0 m/s) should multiply KE by 5² = 25. The graph confirms this: 0.6 J × 25 = 15.0 J, which closely matches the measured 15.2 J. The small difference (0.2 J) is within realistic measurement uncertainty. This verification demonstrates that the v² relationship holds across the full range of the data — KE truly grows with the square of speed.

Tags

S.6.8.A

DOK 2

Blooms: Apply

Relationship of KE to speed

Students in Ms. Patel's class pushed five model cars of different masses along a smooth tabletop so that each car reached a speed of 4 m/s. A motion sensor confirmed each car's speed, and students calculated the kinetic energy.

If Ms. Patel's students repeated the experiment at 8 m/s instead of 4 m/s, would the new graph still show a straight line? Explain how the new graph would compare to the original, using evidence from the graph.

Answer explanation

Claim: The new graph at 8 m/s would still be a straight line through the origin. Evidence: On the 4 m/s graph, each car has a certain amount of kinetic energy at its speed. If the cars go faster, like 8 m/s, their kinetic energy will be higher, but the line will still go straight because heavier cars still have more energy than lighter cars. Reasoning: The graph stays straight because kinetic energy increases evenly with mass at any speed. Increasing speed makes all the energy values bigger, but the pattern — that heavier cars have more energy than lighter cars — stays the same.

Tags

S.6.8.A

DOK 2

Blooms: Apply

Relationship of KE to mass and speed

Students in Ms. Rivera's class attached a 1.5 kg ball to a string and pulley system. They slowly raised the ball to five different heights above the table and calculated the gravitational potential energy at each height.

The ball's GPE at 0.4 m is 5.9 J and at 1.0 m is 14.7 J. Based on the pattern, what would the GPE most likely be at 2.0 m?

• About 16.7 J, because GPE increases by about 2.0 J for each additional meter of height — adding 2.0 J to the 14.7 J at 1.0 m predicts approximately 16.7 J at 2.0 m.

• About 22 J, because the rate of GPE increase slows down at greater heights — the jump from 0.2 to 0.4 m adds 3.0 J, but each later step should add less, giving roughly 22 J at 2.0 m.

• About 58.8 J, because GPE increases with the square of height — from 0.4 to 0.8 m the height doubles and GPE roughly doubles, so from 1.0 to 2.0 m GPE should quadruple.

• About 29.4 J, because the straight-line pattern shows GPE is proportional to height, so doubling the height from 1.0 m doubles the GPE from 14.7 J.

Answer explanation

The graph shows a straight line through the origin, meaning GPE is directly proportional to height. The slope is constant at about 14.7 J/m (which equals mg = 1.5 × 9.8). At 1.0 m the GPE is 14.7 J, so at 2.0 m (double the height) the GPE would be 2 × 14.7 = 29.4 J. This proportional relationship — double the height, double the GPE — is the defining feature of the linear GPE = mgh equation when mass is constant.
A: Uses an incorrect slope value (~2.0 J/m instead of ~14.7 J/m). A student selecting this has misread the graph's rate of increase or confused the x-axis intervals with the y-axis increments. The actual increase from 0.2 m to 0.4 m is about 3.0 J (not 2.0 J), and the pattern is consistent across all intervals.
B: Targets the misconception that GPE increase diminishes at greater heights. The straight-line shape directly contradicts this — a straight line means the rate of increase is constant. A student selecting this may be confusing GPE with a diminishing-returns relationship or expecting a curve that flattens.
C: Targets the misconception that GPE increases with the square of height, overgeneralizing from the KE ∝ v² relationship. GPE = mgh is linear in height, not quadratic. Doubling height doubles GPE, not quadruples it. The straight line on the graph is visual proof of linearity.
D: Correct. The student recognizes the straight-line pattern through the origin as direct proportionality: GPE/height = constant. Doubling from 1.0 m to 2.0 m doubles GPE from 14.7 J to 29.4 J. This extrapolation follows naturally from the linear pattern.

Tags

S.6.8.A

Changes in potential energy due to position/arr...

Blooms: Analyze

DOK 3

Students in Ms. Rivera's class attached a 1.5 kg ball to a string and pulley system. They slowly raised the ball to five different heights above the table and calculated the gravitational potential energy at each height.

The graph shows a straight line from the origin. Using at least two specific data points from the graph, explain why this pattern shows that gravitational PE is directly proportional to height for the 1.5 kg ball.

Answer explanation

Claim: The graph shows that gravitational PE is directly proportional to height for the 1.5 kg ball. Evidence: At 0.4 m, the GPE is 5.9 J. At 0.8 m — double the height — the GPE is 11.8 J, which is double the energy. Similarly, at 0.2 m the GPE is 2.9 J, and at 1.0 m (5× the height) the GPE is 14.7 J (about 5× the energy). Reasoning: In a proportional relationship, multiplying one quantity by a factor multiplies the other by the same factor. Because GPE = mgh and the mass (1.5 kg) and gravity are constant, GPE depends only on height. The constant ratio of GPE to height (about 14.7 J per meter) produces the straight line through the origin, confirming direct proportionality.

Tags

S.6.8.A

DOK 2

Blooms: Apply

Changes in potential energy due to position/arr...

Students in Ms. Okafor's class placed a 0.8 kg textbook on shelves at five different heights in a bookcase. They measured the height of each shelf above the floor and calculated the gravitational PE.

A student says, 'The textbook on the 2.0 m shelf has no gravitational PE because it is just sitting still.' Using at least two data points from the graph, explain why this claim is incorrect.

Answer explanation

Claim: The student's claim is incorrect — the textbook at 2.0 m does have gravitational PE. Evidence: The graph shows the textbook has 15.7 J of GPE at 2.0 m. At 1.0 m it has 7.8 J, and at 2.5 m it has 19.6 J. At every measured height, the stationary textbook has gravitational PE. Reasoning: Gravitational PE depends on an object's height above a reference point, not on whether the object is moving. GPE = mgh means any object with mass at a height above the reference has stored energy due to gravity. The textbook is stationary, but it has the potential to convert this stored energy into kinetic energy if it fell. The student is confusing gravitational PE (which depends on position) with kinetic energy (which depends on motion).

Tags

S.6.8.A

DOK 2

Blooms: Apply

Changes in potential energy due to position/arr...

Students in Ms. Yamada's class used a spring scale and string to slowly lift a 2.0 kg water bottle to five different heights above a desk. They calculated the gravitational PE at each height.

The data points form a straight line that, if extended, would pass through the origin. What does this tell you about the water bottle's GPE at desk level (height = 0 m)?

• The GPE at desk level is negative because the bottle has not been lifted above the desk — below 0.1 m the GPE is 2.0 J, so it must drop below zero.

• The GPE at desk level is zero, because at zero height the bottle has no stored gravitational energy relative to the desk.

• The GPE at desk level is 2.0 J, the same as the lowest data point at 0.1 m — since this is closest to zero height, its value should apply at the desk.

• The GPE at desk level cannot be determined because 0 m was not one of the five heights measured — without a data point at zero, no prediction is possible.

Answer explanation

The straight line through the origin means the y-intercept is zero: when height = 0 m, GPE = 0 J. This makes physical sense — GPE is measured relative to a reference point, and the desk is the reference point in this experiment. An object at the reference height has zero gravitational PE. The proportional relationship (GPE = mgh) confirms this: when h = 0, GPE = m × g × 0 = 0 J, regardless of mass.
A: Targets the misconception that an object at height 0 has negative GPE. GPE at the reference level is zero, not negative. Negative GPE only arises when the object is below the chosen reference point. Since the desk is the reference, height 0 gives GPE = 0.
B: Correct. The student extrapolates the straight line to the origin and connects this to the physical meaning: at the reference height, stored gravitational energy is zero because there is no height above the reference from which the object could fall.
C: Confuses the lowest plotted data point (2.0 J at 0.1 m) with the value at 0 m. The data point at 0.1 m is above the desk, not at the desk. A student selecting this fails to extrapolate the pattern and instead reads the first available point.
D: Targets the misconception that extrapolation from a clear linear pattern is impossible. The straight-line-through-origin relationship makes the prediction at h = 0 straightforward and certain. The student does not need h = 0 to be explicitly measured when the linear pattern clearly indicates GPE = 0 at the origin.

Tags

S.6.8.A

DOK 2

Blooms: Understand

Changes in potential energy due to position/arr...

Students in Ms. Patel's class filled five containers with different amounts of water, creating objects of different masses. They placed each container on a table 1.0 m above the floor and calculated the gravitational PE.

Maya claims the graph proves GPE increases with the square of mass — just like KE increases with the square of speed. Does the graph support Maya's claim?

• Yes, because both GPE and KE depend on squared quantities, so the graph should curve upward — the GPE from 3.9 J at 0.4 kg to 24.5 J at 2.5 kg shows this.

• Yes, because GPE values increase by larger amounts as mass increases — from 3.9 J to 7.8 J to 11.8 J — and these growing jumps show a squared relationship.

• No, because the data forms a straight line through the origin, showing GPE is directly proportional to mass — not proportional to mass squared.

• No, because GPE decreases for the heaviest containers since gravity pulls harder on more massive objects — the 2.5 kg container’s 24.5 J should drop as mass grows.

Answer explanation

Maya's claim confuses the mass-GPE relationship with the speed-KE relationship. KE = ½mv² is quadratic in speed, producing a curve. But GPE = mgh is linear in mass, producing a straight line. The graph shows five data points forming a clear straight line through the origin — at 0.4 kg GPE is 3.9 J, at 0.8 kg it doubles to 7.8 J, at 1.2 kg it triples to 11.8 J relative to 0.4 kg. This constant ratio confirms proportionality, not a squared relationship. A squared relationship would curve upward with increasing steepness, which the graph does not show.
A: Targets the misconception that GPE depends on mass² by analogy with KE depending on v². GPE = mgh has no squared term — mass enters linearly. The student is incorrectly generalizing a pattern from one energy formula to another.
B: Targets a graph-reading error. The raw GPE increases between consecutive points do grow (3.9, 4.0, 5.8, 6.9 J), but only because the mass intervals are unequal (0.4, 0.4, 0.6, 0.7 kg). When normalized per kilogram, the rate is constant at approximately 9.8 J/kg, confirming linearity.
C: Correct. The student must recognize that a straight line through the origin indicates direct proportionality (GPE ∝ mass), which is fundamentally different from the curved pattern that a squared relationship would produce.
D: Targets the misconception that heavier objects have less GPE because gravity pulls them down more. The graph clearly shows GPE increasing with mass. Gravity acting more strongly on heavier objects is what gives them more GPE at a given height, not less.

Tags

S.6.8.A

DOK 2

Blooms: Apply

Relationship of PE to mass

Students in Ms. Kim's class placed five rocks of different masses on a ledge 0.8 m above the ground. They used a balance to measure each rock's mass and calculated the gravitational PE.

A 2.5 kg rock is added to the same ledge. Using the pattern in the graph and at least two existing data points, predict its GPE and explain your reasoning.

Answer explanation

Claim: Based on the graph's linear pattern, a 2.5 kg rock on the 0.8 m ledge would have approximately 19.6 J of gravitational PE. Evidence: The graph shows a straight line through the origin. At 1.0 kg, GPE = 7.8 J, and at 2.0 kg, GPE = 15.7 J. Doubling the mass from 1.0 to 2.0 kg nearly doubles the GPE (15.7 ÷ 7.8 ≈ 2.0). The constant rate of increase is approximately 7.8 J per kilogram. Adding 0.5 kg beyond 2.0 kg should add about 3.9 J (half of 7.8), giving 15.7 + 3.9 ≈ 19.6 J. Reasoning: The straight line through the origin confirms that GPE is directly proportional to mass at constant height. This proportionality means the relationship can be extended beyond the plotted data — each additional kilogram contributes the same ~7.8 J. This rate equals g × h (9.8 × 0.8 = 7.84 J/kg), confirming that the graph reflects the physics of gravitational PE at a fixed height.

Tags

S.6.8.A

Blooms: Analyze

Relationship of PE to mass

DOK 3