D. Energy and Society · Work, Energy and Power

Investigation: Testing Conservation of Energy on a Ramp and a Pendulum

D2.4 D2.3 A1.1 A1.5 A1.6 A1.8 A1.10 A1.13About 75 minutes

Learning goals

I can plan an inquiry to test whether gravitational potential energy is transformed into an equal amount of kinetic energy.
I can identify the independent, dependent and controlled variables and write a hypothesis with a reason.
I can measure a speed from video, calculate Eg and Ek, and compare them with a percentage.
I can analyse sources of error and draw a conclusion supported by my data.

Words to know

hypothesis
A testable prediction with a reason, often in the form "If ..., then ..., because ...".

independent variable
The variable the experimenter changes on purpose (here, the release height).

dependent variable
The variable that is measured to see the effect (here, the speed at the bottom).

controlled variables
Variables kept the same so the test is fair (mass of the car, the ramp, the surface, how the car is released).

percent difference / percent error
Ways to compare a measured value with an expected one: |expected − measured| ÷ expected × 100%.

systematic error
An error that pushes every result in the same direction, such as friction always making the speed lower.

random error
Unpredictable scatter in measurements, such as small differences in timing from one trial to the next. Repeating trials reduces its effect.

Don't just believe it: test it

You have used Ek₁ + Eg₁ = Ek₂ + Eg₂ to solve problems. But how do we know that the gravitational potential energy an object loses really becomes the same amount of kinetic energy? In science, laws are trusted because they have survived many careful tests. In this lesson you will plan and carry out your own test, using a ramp (or a pendulum), a ruler and a phone camera, and then judge your results like a scientist.

Planning the inquiry

Question: When a toy car rolls down a ramp, is the kinetic energy it gains equal to the gravitational potential energy it loses?

Hypothesis: If the release height is increased, then the speed at the bottom will increase so that ½mv² is close to mgh, but slightly less, because energy is conserved and a small amount will become thermal energy through friction and air resistance.

Variables:

Independent: release height h (0.10 m, 0.20 m, 0.30 m, 0.40 m)
Dependent: speed v on the flat track just after the ramp
Controlled: the same car (same mass), the same ramp and surface, releasing from rest (no push), the same timing zone

Measuring speed: film the car crossing a marked 0.50 m timing zone on the flat section. Slow-motion video (often 120 or 240 frames per second) lets you count frames: time = (number of frames) ÷ (frames per second). Then v = 0.50 m ÷ time. In a school lab you could use a photogate or a motion sensor instead.

Repeat each height at least three times and average, to reduce random error.

The ramp set-up. Measure h from the flat track to the bottom of the car's wheels at its starting point. The phone films the car crossing the 0.50 m timing zone. (Not to scale.)

Worked example

Example 1: Analysing one trial

A 0.040 kg toy car is released from rest at h = 0.20 m. It crosses the 0.50 m timing zone in 0.27 s. Compare the Ek gained with the Eg lost.

Speed: v = 0.50 m ÷ 0.27 s ≈ 1.85 m/s.
Eg lost: ΔEg = mgh = (0.040)(9.8)(0.20) = 0.0784 J.
Ek gained: Ek = ½(0.040)(1.85)² ≈ 0.0686 J. (Keep extra digits until the end.)
Percentage of Eg that became Ek: 0.0686 ÷ 0.0784 × 100% ≈ 87.5%.
Predicted speed with no friction: v = √(2gh) = √(2 × 9.8 × 0.20) ≈ 1.98 m/s. Percent difference in speed: (1.98 − 1.85) ÷ 1.98 × 100% ≈ 6.5%.

About 88% of the lost Eg appeared as Ek. The measured speed is about 6.5% lower than the no-friction prediction.

h (m)	average time (s)	v (m/s)	ΔEg = mgh (J)	Ek = ½mv² (J)	Ek ÷ ΔEg
0.10	0.38	1.32	0.0392	0.0346	88%
0.20	0.27	1.85	0.0784	0.0686	88%
0.30	0.22	2.27	0.1176	0.1033	88%
0.40	0.19	2.63	0.1568	0.1385	88%

Sample results for a 0.040 kg car. Times are averages of three trials, read from 240 frames/s video.

Graphing the sample results. The points lie on a straight line through the origin, but its slope is about 0.88, below the dashed 100% line: about 88% of the lost Eg became Ek every time.

Analysing the results

Pattern. Ek grows in proportion to ΔEg (a straight line through the origin). That supports the idea that height energy is being turned into motion energy.

Gap. Ek is consistently about 12% less than ΔEg. Because the gap is in the same direction every time, it points to a systematic cause, not random error:

Friction in the axles and between the wheels and the ramp, plus air resistance, turn some energy into thermal energy (and sound).
The spinning wheels also carry some kinetic energy of rotation that our ½mv² does not count.
The car slows slightly while crossing the timing zone, so the average speed there is a little lower than the speed at the bottom of the ramp.

Random errors include reading the frame where the car crosses each mark (±1 frame at 240 frames/s is ±0.004 s), and measuring h to the nearest millimetre. Repeating trials and averaging reduces their effect.

Significant digits. Heights were measured to 2 significant digits and times to 2, so final answers should be given to about 2 significant digits (for example, 88%, 1.9 m/s).

Worked example

Example 2: Writing the conclusion

Write a conclusion for the ramp investigation, using the sample results.

Answer the question directly: the kinetic energy gained was close to, but less than, the gravitational potential energy lost.
Use evidence: at every height, Ek was about 88% of ΔEg, and a graph of Ek against ΔEg was a straight line through the origin with slope 0.88.
Explain the difference: about 12% of the energy became thermal energy and sound through friction and air resistance, and some was rotational energy of the wheels.
Link to the law: the results support the law of conservation of energy, because the "missing" energy is accounted for by forms we did not measure.
Suggest an improvement: use a low-friction cart with light wheels, or measure the speed closer to the bottom of the ramp.

Sample conclusion: "The car's kinetic energy at the bottom was about 88% of the gravitational potential energy it lost, at all four heights. This supports conservation of energy: the remaining 12% was most likely transformed into thermal energy and sound by friction and air resistance, and into rotational energy of the wheels. A lower-friction cart would test this further."

The pendulum version

No ramp? Use a pendulum. Hang a heavy nut on 1 m of string from a doorway. Release it from a measured height above its lowest point and film the opposite side to find how high it rises. Conservation predicts it rises to almost the same height. You can also find the speed at the bottom by filming the bob as it passes a 5 cm mark centred on its lowest point; there the speed changes very little, so v ≈ 0.05 m ÷ time.

Just for fun

Lab motto: if at first you don't succeed, call it Trial 1.

Hands-on activity

Run the ramp investigation

You need: a toy car (or a small ball), a smooth board or a stiff binder about 60 cm long, books to prop it up, a metre stick, masking tape, a kitchen scale and a phone with slow-motion video.

Grown-up note: prop the ramp securely on the floor so books cannot fall, and keep the area clear for walking.

Weigh the car (in kg). Prop the ramp so it ends smoothly on a hard, flat floor.
Use tape to mark a starting line on the ramp, and two marks 0.50 m apart on the floor just past the bottom of the ramp.
Measure the starting height h. Release the car from rest. Film it crossing the timing zone. Do three trials.
Repeat for three more heights (raise the start line, or add books).
Count frames to find each time, then calculate v, ΔEg, Ek and Ek ÷ ΔEg. Graph Ek against ΔEg.
Write a conclusion and two sources of error, saying whether each is random or systematic.

What to look for: a straight-line graph with a slope a bit less than 1. Toy cars often give 70% to 95%; balls that roll lose more because a large share of their energy goes into spinning. A result above 100% means a measurement mistake (for example, h measured to the wrong point or the car was pushed).

Your turn

Write a hypothesis for this question: "How does the release height of a pendulum affect its speed at the bottom?"

Identify the independent, dependent and two controlled variables in the ramp investigation.

A 0.060 kg cart released from 0.25 m crosses a 0.50 m zone in 0.24 s. Find v, ΔEg, Ek and the percentage of Eg that became Ek.

At 240 frames per second, a car crosses a timing zone in 72 frames. What is the time?

Every trial in an experiment gives an Ek about 10% lower than the ΔEg. Is this more likely a random or a systematic error? Suggest a cause.

Why should each trial be repeated and averaged?

More practice: warm-up

What does the independent variable mean?

Write the formula for percent difference between an expected and a measured value.

A predicted speed is 2.0 m/s and the measured speed is 1.9 m/s. Find the percent difference.

Name one tool used in a school lab to measure speed accurately.

Why must the car be released from rest, not pushed?

More practice: on your own

A student finds Ek = 0.21 J and ΔEg = 0.20 J. What does this suggest, and what should the student check?

A pendulum is released 0.12 m above its lowest point. On the other side, it rises to 0.11 m. What percentage of its mechanical energy did it keep over half a swing? Where did the rest go?

Explain why the time read from a video at 30 frames per second is less precise than at 240 frames per second.

A ball rolls (not slides) down a ramp. Its Ek (½mv²) is only about 70% of ΔEg, even with very little friction. Suggest a reason.

Using the sample data table, what would you predict for Ek at h = 0.50 m? Show your reasoning.

Challenge

Using the sample data at h = 0.40 m, find the average friction force on the car, assuming all the missing energy went into friction and the car travelled 0.80 m along the ramp.

Design a change to the investigation that would test whether air resistance is a major cause of the energy difference. Name your variables.

The time for one run is measured as 0.20 s ± 0.01 s over a 0.50 m zone. Find the range of possible speeds and the range of Ek for a 0.040 kg car. What does this tell you about the precision of your Ek values?

Check your understanding

Quiz

In an investigation testing conservation of energy on a ramp, the independent variable is best chosen as:

Results show Ek is always about 85% of ΔEg. This is most likely due to:

A car crosses a 0.60 m timing zone in 0.30 s. Its speed is:

A graph of Ek (vertical) against ΔEg (horizontal) for perfect energy conservation would be:

The predicted speed is 3.0 m/s and the measured speed is 2.7 m/s. The percent difference is:

Which change would reduce random error in the measured speed?

For parents and teachers

Big idea: This lesson is the D2.4 investigation, and it practises the scientific investigation skills of strand A: hypotheses and variables (A1.1), controlled experiments (A1.5), organizing data (A1.6), error analysis (A1.8), conclusions (A1.10) and significant digits (A1.13).

How to help: Be the lab partner: hold the phone, release the car, or read frames. Ask "How do you know?" when your teen writes a conclusion, and look for numbers as evidence.

Common difficulties:

Measuring h to the top of the ramp instead of the car's actual starting height.
Pushing the car instead of releasing it.
Rounding too early (keep extra digits until the end).
Treating a result over 100% as a discovery rather than a measurement mistake.

Safety: secure the ramp so books cannot fall, keep the floor clear, and keep small toy parts away from young children.