Scientific theories

Where the answer comes from: the passage says an object "floats in water if its density is less than the density of water, 1.00 g/cm³." The table gives each object's mass and volume.

Step by step:

1. Object A: d = \dfrac{54.0}{20.0} = 2.70 g/cm³. Greater than 1.00 → sinks.
2. Object B: d = \dfrac{18.0}{24.0} = 0.75 g/cm³. Less than 1.00 → floats.
3. Object C: d = \dfrac{89.6}{10.0} = 8.96 g/cm³. Greater than 1.00 → sinks.

Correct answer: B (Object B only).

Notice that Object B has the largest volume but the smallest mass. Floating depends on density (mass per volume), not on size or mass alone.

Tip: Applying a scientific rule means computing the quantity the rule is about (here, density) and then comparing it with the rule's cutoff (1.00 g/cm³).

Where the answer comes from: the table gives Object C a mass of 89.6 g and a volume of 10.0 cm³. The reference table lists iron at 7.87 g/cm³ and copper at 8.96 g/cm³.

Step by step:

1. d = \dfrac{m}{V} = \dfrac{89.6\text{ g}}{10.0\text{ cm}^3} = 8.96 g/cm³.
2. Compare with the reference table: copper is exactly 8.96 g/cm³.
3. Iron (7.87) is close but does not match; aluminum (2.70) matches Object A instead.
4. The density can be determined because both the mass and the volume are given.

Correct answer: B (copper).

Tip: Density is a characteristic property: a pure substance has the same density no matter the size of the sample. That is why it can identify materials.

Where the answer comes from: the passage says the water rose from 15.0 mL to 45.0 mL and "The volume of the sample equals the rise in the water level." The mass was 79.5 g.

Step by step:

1. Volume: V = 45.0 - 15.0 = 30.0 mL = 30.0 cm³.
2. Density: d = \dfrac{m}{V} = \dfrac{79.5\text{ g}}{30.0\text{ cm}^3} = 2.65 g/cm³.
3. Match the table: 2.65 g/cm³ is quartz.

Correct answer: A (quartz).

Why the others are wrong: Hematite (5.3) is what you get if you divide by the starting reading, 15.0 mL, instead of the change. Calcite (2.71) is close but does not match 2.65. Galena (7.6) is far too dense.

Check: 2.65 \times 30.0 = 79.5 g ✓

Tip: In water displacement, the object's volume is the difference between two readings, never one reading by itself.

Where the answer comes from: the passage says about "1 in every 1,000,000 bacteria carries a random gene mutation that lets it survive" the antibiotic, and "The other bacteria are killed by the drug." It also says natural selection "acts on variation that already exists."

Step by step:

1. Variation: a few bacteria already have the resistance mutation before the drug is taken.
2. Selection: the drug kills mostly the bacteria without the mutation.
3. Inheritance: survivors pass the resistance gene to their offspring.
4. Result: when the population regrows, a much larger fraction of it is resistant.

Correct answer: B.

Why the others are wrong: A describes the drug creating resistance on demand, which the passage says natural selection does not do. C ignores that the survivors were mostly resistant ones. D reverses the effect: the drug removed the non-resistant bacteria.

Tip: In natural selection questions, follow the three steps: variation already present, the environment removes some, survivors' traits become more common.

Where the answer comes from: the passage says the pronghorn "ran 800 meters in 32 seconds" and gives v = \dfrac{d}{t}.

Step by step:

1. Substitute: v = \dfrac{800\text{ m}}{32\text{ s}}.
2. Divide: 800 \div 32 = 25.
3. Units: meters ÷ seconds = meters per second.

Correct answer: C (25 m/s).

Check: 25\text{ m/s} \times 32\text{ s} = 800 m ✓

Tip: If you get a tiny number like 0.04 for an animal's speed, you probably divided time by distance. The unit "m/s" reminds you: meters on top, seconds on the bottom.

Where the answer comes from: the passage gives Newton's second law, F = ma, with force in newtons, mass in kilograms and acceleration in m/s².

Step by step:

1. Write the formula: F = ma.
2. Substitute: F = 5.0\text{ kg} \times 2.0\text{ m/s}^2.
3. Multiply: 5.0 \times 2.0 = 10.
4. A kg·m/s² is a newton, so F = 10 N.

Correct answer: C (10 N).

Check with the table: Trial 1 had 2.0 \times 3.0 = 6.0 N, matching the measured force, so the law fits the students' data.

Tip: When a formula is given, list what you know (m = 5.0, a = 2.0) and what you need (F) before substituting. That prevents dividing when you should multiply.

Where the answer comes from: the table gives Earth to the Sun as 1.5 \times 10^8 km, and the passage gives the speed of light as 3.0 \times 10^5 km/s.

Step by step:

1. Formula: t = \dfrac{d}{v} = \dfrac{1.5 \times 10^8\text{ km}}{3.0 \times 10^5\text{ km/s}}.
2. Divide the front numbers: 1.5 \div 3.0 = 0.5.
3. Divide the powers of ten by subtracting exponents: 10^{8-5} = 10^3.
4. Combine: 0.5 \times 10^3 = 5.0 \times 10^2 s, which is 500 seconds.
5. Units: km ÷ (km/s) = s.

Correct answer: A (5.0 \times 10^2 seconds). That is 500 \div 60 \approx 8.3 minutes.

Tip: When dividing in scientific notation, subtract exponents; when multiplying, add them. Then adjust so the front number is between 1 and 10.

Where the answer comes from: the passage says air could flow into the swan-neck flask, but "dust and microbes from the air settled in the bend of the neck and could not reach the broth." Flask 1 "stayed clear for many weeks," while Flask 2, with the neck broken off, "became cloudy with microbes within days."

Step by step:

1. Spontaneous generation predicts microbes would appear in any broth exposed to air.
2. Flask 1 had air but no incoming microbes, and no life appeared.
3. Flask 2 let microbes fall in, and microbes then grew.
4. So microbes appeared only when existing microbes could get in, which supports "all cells come from pre-existing cells."

Correct answer: C.

Why the others are wrong: A is part of the method, not a result. B is true but, by itself, it could still fit spontaneous generation; the key is the comparison with Flask 1. D is false: the passage says air could reach the broth.

Tip: Evidence for a theory is strongest when it comes from a comparison: two setups that differ in one way, with different results.

Where the answer comes from: the passage gives KE = \dfrac{1}{2}mv^2, says "the braking distance is proportional to the kinetic energy," and reports a braking distance of 18 m at 40 km/h.

Step by step:

1. Speed doubles: 40 km/h → 80 km/h, so v becomes 2v.
2. Kinetic energy depends on v^2: (2v)^2 = 4v^2, so the kinetic energy becomes 4 times as large.
3. Braking distance is proportional to kinetic energy, so it also becomes 4 times as large.
4. 4 \times 18\text{ m} = 72 m.

Correct answer: C (72 m).

Why the others are wrong: 36 m assumes distance simply doubles with speed, ignoring the square. 18 m assumes speed does not matter. 144 m multiplies by 8, as if speed were cubed.

Check: ratio of speeds squared: \left(\dfrac{80}{40}\right)^2 = 4 ✓

Tip: When a variable is squared, doubling it multiplies the result by 4 and tripling it multiplies by 9. Look for exponents before you assume 'double in, double out.'

Where the answer comes from: the passage gives Q = mc\Delta T, the block's mass (50.0 g), the energy delivered (900 J), and its temperatures (22.0 °C to 42.0 °C). The table lists specific heats.

Step by step:

1. Temperature change: \Delta T = 42.0 - 22.0 = 20.0 °C.
2. Rearrange for c: c = \dfrac{Q}{m\Delta T}.
3. Substitute: c = \dfrac{900}{50.0 \times 20.0} = \dfrac{900}{1{,}000} = 0.90 J/(g °C).
4. Match the table: 0.90 is aluminum.

Correct answer: A (aluminum).

Why the others are wrong: Iron (0.45) is close to what you get if you use the final temperature, 42.0 °C, instead of the change: 900 \div (50.0 \times 42.0) \approx 0.43. Copper (0.39) and lead (0.13) do not match any correct setup.

Check: 50.0 \times 0.90 \times 20.0 = 900 J ✓

Tip: Read the definitions under a formula. Here the passage says \Delta T is the change in temperature; using a single thermometer reading is the most common error.

Where the answer comes from: the passage gives Ohm's law, V = IR, a 9.0 V battery, and in the question a 30 Ω resistor.

Step by step:

1. Step 1 is correct: it is Ohm's law.
2. To get I alone, divide both sides by R: I = \dfrac{V}{R}. Step 2 multiplied instead, so Step 2 is the error.
3. Correct calculation: I = \dfrac{9.0\text{ V}}{30\ \Omega} = 0.30 A.

Correct answer: B.

Why the others are wrong: A is wrong because 270 A is the result of the faulty Step 2 (a 9 V battery could never push 270 A through a 30 Ω resistor). C inverts the formula (R \div V). D blames the arithmetic, but 9.0 \times 30 really is 270; the setup is what is wrong.

Check: 0.30\text{ A} \times 30\ \Omega = 9.0 V ✓

Tip: After you rearrange a formula, plug your answer back into the original formula. If it does not give the known value, the rearranging went wrong.

Where the answer comes from: the table gives Earth to Mars at closest approach as 5.6 \times 10^7 km; the speed of light and radio signals is 3.0 \times 10^5 km/s; 1 minute = 60 seconds.

Step by step:

1. One-way time: t = \dfrac{5.6 \times 10^7}{3.0 \times 10^5} = \dfrac{5.6}{3.0} \times 10^{7-5} \approx 1.87 \times 10^2 \approx 187 s.
2. Round trip (signal there and reply back): 2 \times 187 \approx 373 s.
3. Convert to minutes: 373 \div 60 \approx 6.2 minutes.

Correct answer: C (about 6.2 minutes).

Why the others are wrong: 3.1 minutes is only the one-way trip. 373 is the round-trip time in seconds, not minutes. 62 minutes comes from a decimal-place slip when converting.

Tip: In multi-step problems, write the units at every step. If your final unit is not what the question asks for (here, minutes), you are not done.

Where the answer comes from: the passage states the law: "energy cannot be created or destroyed; it can only change from one form to another." It also says that the prediction assumed "no energy is changed into other forms." The measured speed was 18.5 m/s.

Step by step:

1. Predicted KE with no losses: 98 J (v \approx 19.8 m/s).
2. Measured KE: \tfrac{1}{2}(0.50)(18.5)^2 = 0.25 \times 342.25 \approx 85.6 J.
3. About 98 - 85.6 \approx 12 J of the starting energy did not become kinetic energy.
4. By the law, those joules were not destroyed. As the ball pushed through the air, air resistance changed some energy into thermal energy (slight heating of the ball and air) and sound. Total energy is still conserved.

Correct answer: B. A contradicts the law. C is false — the law applies to all objects. D is unsupported: the ball's mass does not change during the fall, and extra mass would not lower the speed in this way.

Tip: When a measurement falls short of a theory's "ideal" prediction, look for a condition the ideal case ignored (friction, air resistance, heat loss) rather than concluding the law is wrong.

Where the answer comes from: for 1.0 N to 4.0 N the stretch is exactly 2.5 cm per newton (k = 0.40 N/cm), so F = kx predicts x = \dfrac{6.0}{0.40} = 15.0 cm at 6.0 N. The passage says that beyond the elastic limit "the spring stretches more than the law predicts, and it does not return to its original length." After the 6.0 N weight, the spring "stayed 3 cm longer."

Step by step:

1. The 6.0 N stretch (21.0 cm) is more than predicted, as the passage says happens past the elastic limit.
2. The spring was permanently lengthened, the second sign of passing the elastic limit.
3. So Hooke's law still describes the spring below its limit; it simply does not apply above it.

Correct answer: D.

Why the others are wrong: A is too absolute; the first five rows fit the law perfectly. B would cause less stretch, not more. C has no support: random reading errors would not also leave the spring permanently longer.

Tip: Every law has conditions. When data break a law's prediction, first ask whether the experiment went outside those conditions.

Where the answer comes from: the passage defines Q = mc\Delta T, where \Delta T is "the change in temperature (final temperature minus starting temperature)," and lists copper's specific heat as 0.39 J/(g °C).

Step by step:

1. Step 1 correctly finds \Delta T = 15.0 °C.
2. Step 2 should use that change, but it uses 35.0, the final temperature. This is the error.
3. Correct calculation: Q = 200 \times 0.39 \times 15.0 = 1{,}170 J.
4. Step 3's arithmetic (200 \times 0.39 \times 35.0 = 2{,}730) is correct for the wrong setup.

Correct answer: C.

Why the others are wrong: A is wrong because Step 1 is correct (final minus start). B accepts an answer built on the wrong temperature. D blames the arithmetic, which is actually correct.

Check: \dfrac{1{,}170}{200 \times 0.39} = 15.0 °C ✓

Tip: When you are asked to find an error, check each step against the formula's definitions, not only the arithmetic.

Where the answer comes from: the bulb measured 12 Ω "at room temperature, before it was switched on," and drew 0.20 A at 6.0 V while glowing. The teacher's note says resistance "rises as the metal gets hotter" and a filament "can be well over 1,000 °C."

Step by step:

1. Prediction with the cold resistance: I = \dfrac{6.0}{12} = 0.50 A.
2. Actual resistance while glowing: R = \dfrac{V}{I} = \dfrac{6.0}{0.20} = 30 Ω.
3. The filament's resistance rose from 12 Ω to 30 Ω as it heated, so less current flowed.
4. Ohm's law still holds; it just must use the hot resistance.

Correct answer: A.

Why the others are wrong: B is too absolute; Ohm's law works with the resistance the bulb actually has. C misunderstands current: current is not used up inside a bulb. D points the wrong way: more voltage would push more current.

Tip: When a measurement disagrees with a law, check whether a quantity the law treats as fixed (here, resistance) actually changed during the experiment.

Where the answer comes from: the passage says the force is "directly proportional to each mass" ("Doubling one mass doubles the force") and "inversely proportional to the square of the distance" ("Doubling the distance makes the force \tfrac{1}{2^2} = \tfrac{1}{4} as large").

Step by step:

1. Mass doubles: force × 2.
2. Distance doubles: force × \dfrac{1}{4}.
3. Combine: F \times 2 \times \dfrac{1}{4} = \dfrac{2}{4}F = \dfrac{1}{2}F.

Correct answer: B (\tfrac{1}{2}F).

Why the others are wrong: A assumes the two doublings cancel, ignoring that distance is squared. C applies only the distance change. D applies only the mass change.

Check with the formula: \dfrac{G(2m_1)m_2}{(2d)^2} = \dfrac{2}{4} \times \dfrac{Gm_1m_2}{d^2} ✓

Tip: When two variables change at once, write the factor for each change and multiply the factors together.

Where the answer comes from: the passage gives weight = mg with g = 9.8 m/s², W = Fd, and P = \dfrac{W}{t}, where "t is time in seconds."

Step by step:

1. Force needed = weight: 60 \times 9.8 N.
2. Work = force × height: 60 \times 9.8 \times 4.0 J.
3. Convert time: 0.25\text{ min} \times 60\ \dfrac{\text{s}}{\text{min}} = 15 s.
4. Power: P = \dfrac{60 \times 9.8 \times 4.0}{15} \approx 157 W.

Correct answer: C.

Why the others are wrong: A uses the mass (60 kg) as if it were a force. B forgets to convert 0.25 minute to seconds. D multiplies by time instead of dividing.

Check: C gives 2{,}352 \div 15 \approx 157 W, a reasonable value for a person climbing stairs quickly.

Tip: On "which expression" items, check three things in order: the right formula, the right quantity in each slot (force, not mass), and the right units (seconds, not minutes).

Where the answer comes from: the passage says redshifts show distant galaxies "are moving away from us," that "The farther away a galaxy is, the faster it moves away," and that the Big Bang theory explains this by expansion. It also says a few "very close galaxies, such as the Andromeda Galaxy, are moving toward us" because of gravity, and "The expansion pattern shows up over large distances."

Evaluate each option:

• A fits the theory: the passage already explains nearby galaxies moving toward us as a local gravity effect.
• B fits Hubble's law: twice the distance, about twice the speed.
• C is the cosmic microwave background, which the passage lists as support for the theory.
• D breaks the large-scale pattern. If most distant galaxies showed blueshifts (moving toward us), the universe would not be expanding as the theory claims.

Correct answer: D.

Tip: Evidence challenges a theory only if it contradicts the theory's main prediction. A known, explained exception (like Andromeda) does not overturn a broad pattern.

Where the answer comes from: the passage gives \text{rate} = \dfrac{\text{distance}}{\text{time}} and "1 km = 100,000 cm." The question gives a rate of 8.0 cm/yr and a width of 4,000 km.

Step by step:

1. Rearrange for time: \text{time} = \dfrac{\text{distance}}{\text{rate}}.
2. Convert distance: 4{,}000\text{ km} = 4.0 \times 10^3 \times 1.0 \times 10^5\text{ cm} = 4.0 \times 10^8 cm.
3. Divide: \dfrac{4.0 \times 10^8\text{ cm}}{8.0\text{ cm/yr}} = 0.50 \times 10^8 = 5.0 \times 10^7 yr.

Correct answer: A (5.0 \times 10^7 years, or 50 million years).

Why the others are wrong: B divides 4,000 by 8.0 without converting km to cm. C converts only to meters (4.0 \times 10^6 m) while the rate is in cm. D multiplies distance by rate instead of dividing.

Check: 5.0 \times 10^7\text{ yr} \times 8.0\text{ cm/yr} = 4.0 \times 10^8 cm = 4{,}000 km ✓

Tip: Rewrite each quantity in the same unit and in scientific notation before dividing; then divide the front numbers and subtract the exponents.