A1.8 Analyse evidence, solve quantitative problems, and evaluate error

Ontario Grade 12 Physics

Scientific Investigation Skills and Career Exploration

A practical framework for analysing physics results at the Grade 12 level

In SPH3U, you used measurements and models to describe motion, forces, energy, waves, circuits, and magnetism. A physics result is stronger when you show how evidence supports it, explain the relationship used, and consider measurement error. In this lesson, evidence means observations or measurements that can be used to assess a physics claim. Error is the difference between a measured or calculated result and a reference value, or a limitation that affects the result. Error does not automatically mean that someone made a mistake. The aim is to solve carefully and judge what the evidence can support.

What you will learn

Identify the system, reference frame, positive direction, and quantities in a physics problem.
Use evidence and a suitable physics relationship to solve quantitative problems.
Report answers with SI units, appropriate significant figures, and vector direction when needed.
Distinguish measurement uncertainty, random error, and systematic error.
Evaluate whether an answer is reasonable and explain how error may affect a result.

1. Set up the system and read the evidence

A physical system is the object or group of objects being studied. A reference frame is the viewpoint used to describe position and motion. Before calculating, state what the system is and what frame applies. For a cart moving along a track, the cart can be the system and the track can define the reference frame.

Choose a positive direction and keep it consistent. For example, let right along the track be positive. Displacement and velocity are vectors: they have both magnitude and direction. Time, mass, and speed are scalars: they have magnitude but no direction. A negative vector component means the quantity points opposite to the chosen positive direction.

Read the evidence before selecting an equation. Record known values with units, identify the unknown, and note any stated uncertainty. An uncertainty is an estimated range around a measured value. If a length is recorded as

0.420 \pm 0.002\ \mathrm{m}

, the measurement is reported as

0.420\ \mathrm{m}

with an estimated uncertainty of

0.002\ \mathrm{m}

.

State the system, reference frame, and positive direction.
Separate measured evidence from assumptions or a proposed procedure.
Keep vector direction visible through signs or words.

2. Choose a relationship and solve with units

A physics relationship links quantities in a model. Choose one that matches the evidence and the conditions in the question. For example, average velocity is displacement divided by elapsed time. Newton’s second law relates net force to mass and acceleration. A relationship is useful only when its quantities represent the situation being studied.

Rearrange the relationship before substituting if that makes the unknown clear. Substitute values with units, not just numbers. Keep track of signs for vector quantities. For a calculation with measured values, use a sensible number of significant figures. Significant figures are the digits that communicate the precision of a reported value. Do not report more detail than the evidence supports.

After calculating, check units and reasonableness. The units should follow from the relationship. Ask whether the size and direction fit the situation. A cart moving in the negative direction should not receive a positive velocity unless the chosen direction or evidence justifies it.

v_{\mathrm{avg}}=\frac{\Delta x}{\Delta t}

Write the governing relationship before substitution.
Carry units through the calculation.
Check the answer’s units, sign, direction, and reasonable size.

3. Describe and evaluate error

Random error causes repeated measurements to vary unpredictably. It can come from small differences in timing or reading a scale. Repeating measurements and comparing them can help show this variation. Systematic error shifts measurements in a consistent way. A miscalibrated sensor is one possible cause. Repeating measurements alone may not remove a systematic shift.

Accuracy describes how close a result is to an accepted or reference value. Precision describes how closely repeated results agree with one another. A set of results can be precise but not accurate if a consistent bias affects every reading. Measurement uncertainty describes the estimated limits of a measurement; it is not the same as a known error.

When a reference value is provided, compare the result with it. Percent error expresses the size of the difference relative to the reference value. Use absolute values when reporting the size of the error. The signed difference can still help identify whether a result is higher or lower than the reference. Do not claim that a particular source caused an error unless the evidence supports that explanation.

When evaluating a result, identify a plausible source of error, explain how it would affect the measured quantity, and suggest a relevant improvement. For example, if reaction time affects a stopwatch reading, using a timing method that reduces the effect of reaction time may improve the evidence. Keep actual measurements distinct from practice values, estimates, or proposed procedures.

\%\,\mathrm{error}=\frac{|\mathrm{measured}-\mathrm{reference}|}{|\mathrm{reference}|}\times100\%

Random error varies; systematic error tends to shift results consistently.
Accuracy and precision describe different qualities of results.
An error evaluation should connect a possible source to its effect and a practical improvement.

4. Make the evidence-based conclusion

A conclusion should answer the question using the calculated result and the evidence provided. State the result with units and direction where relevant. Then explain whether the result is consistent with the expected relationship or reference value, considering the stated uncertainty or likely limitations.

Avoid claiming more than the evidence shows. If two values differ slightly, that does not by itself prove a specific cause. If uncertainty ranges overlap, the measurements may be consistent within their stated limits, but that does not prove the quantities are exactly equal. A clear conclusion separates what was observed, what was calculated, and what is being inferred.

Use evidence to support a conclusion, not just a formula.
State limits and uncertainty without inventing causes or measurements.
Distinguish an observation from an interpretation.

Worked example

Finding average velocity from position evidence

Practice data, not a reported laboratory result: a cart moves from

x_i=0.120\ \mathrm{m}

to

x_f=0.780\ \mathrm{m}

in

2.0\ \mathrm{s}

. The positive direction is to the right along the track. Find its average velocity.

Define the motion
The system is the cart, and the track is the reference frame. Right is positive. The unknown is average velocity. The positions and elapsed time are the given evidence.
Find displacement
Displacement is final position minus initial position. The positive result means the cart’s net change in position is to the right.
$\Delta x=x_f-x_i=0.780\ \mathrm{m}-0.120\ \mathrm{m}=0.660\ \mathrm{m}$
Calculate average velocity
Use displacement divided by elapsed time. The units reduce to metres per second, and the positive sign indicates motion in the chosen positive direction.
$v_{\mathrm{avg}}=\frac{0.660\ \mathrm{m}}{2.0\ \mathrm{s}}=0.33\ \mathrm{m/s}$

Answer: The cart’s average velocity is

0.33\ \mathrm{m/s}

to the right.

Check: The units are metres per second. The displacement is positive, so the direction agrees with the sign convention. The average speed is modest for a cart travelling less than a metre in two seconds.

Worked example

Using net force evidence

Practice data: a

2.0\ \mathrm{kg}

cart has a net force of

3.6\ \mathrm{N}

to the right. Find its acceleration. Take right as positive.

Define the system and direction
The system is the cart, described relative to the track. Right is positive. Mass and net force are given; acceleration is the unknown. Force and acceleration are vectors, while mass is a scalar.
Select the relationship
Newton’s second law connects net force, mass, and acceleration. The net force is already given, so use it directly rather than adding individual forces again.
$F_{\mathrm{net}}=ma$
Substitute and solve
Rearrange for acceleration and substitute the force and mass with SI units. Since the net force is positive, the acceleration is in the positive direction.
$a=\frac{F_{\mathrm{net}}}{m}=\frac{+3.6\ \mathrm{N}}{2.0\ \mathrm{kg}}=+1.8\ \mathrm{m/s^2}$

Answer: The cart’s acceleration is

1.8\ \mathrm{m/s^2}

to the right.

Check: A newton is equivalent to a kilogram metre per second squared, so force divided by mass gives acceleration units. A positive net force produces positive acceleration in this model. The result is reasonable for the stated force on a small mass.

Worked example

Evaluating error against a reference value

Practice calculation: a measured wave speed is

19.2\ \mathrm{m/s}

. A reference value supplied for comparison is

20.0\ \mathrm{m/s}

. Calculate the percent error and interpret it.

Identify comparison values
Treat the wave and its medium as the system, and use the same reference frame for both speeds. Speed is a scalar. The measured value is $19.2\ \mathrm{m/s}$ , and the supplied reference is $20.0\ \mathrm{m/s}$ .
Find the magnitude of the difference
Subtract the reference value from the measured value. The signed difference is negative, so the measured speed is lower; use its magnitude to calculate percent error.
$|19.2\ \mathrm{m/s}-20.0\ \mathrm{m/s}|=0.8\ \mathrm{m/s}$
Calculate percent error
Divide the difference by the magnitude of the reference and multiply by 100%. Report the result to two significant figures, consistent with the input values.
$\%\,\mathrm{error}=\frac{0.8\ \mathrm{m/s}}{20.0\ \mathrm{m/s}}\times100\%=4.0\%$

Answer: The percent error is 4.0%. The measured speed is below the reference value.

Check: The units cancel in the ratio, leaving a percent. The difference is less than the reference, so an error of a few percent is reasonable. This comparison alone does not identify the cause of the difference.

Common mistakes and how to avoid them

Reporting a vector answer as a positive number without a direction.

Correction: Include a direction word or a signed component tied to the stated positive direction.

Treating uncertainty as proof that a measurement is wrong.

Correction: Uncertainty is an estimated range. Use it to describe the limits of the evidence, not to assert a known mistake.

Calling a result accurate just because repeated results are close together.

Correction: Close agreement indicates precision. Accuracy requires comparison with an accepted or reference value.

Naming a specific error source without evidence.

Correction: Describe it as a possible source and explain how it could affect the result.

Lesson summary

Define the system, reference frame, positive direction, known values, and unknown.
Use a physics relationship that matches the evidence and conditions.
Carry SI units through the calculation and report sensible significant figures.
Check units, direction, sign, and reasonableness.
Evaluate error by comparing results carefully and explaining limitations supported by evidence.

Check your understanding

Question 1

A motion calculation gives

-2.4\ \mathrm{m/s}

when right is positive. What does the sign mean?

The object moves left at $2.4\ \mathrm{m/s}$ .
The object moves right at $2.4\ \mathrm{m/s}$ .
The object has no motion because speed cannot be negative.
correctIndex eccentricity

Show answer and explanation

The object moves left at

2.4\ \mathrm{m/s}

.

The negative sign means the velocity points opposite to the chosen positive direction, so it is to the left.

Question 2

A reference speed is

10.0\ \mathrm{m/s}

and a measured speed is

9.0\ \mathrm{m/s}

. What is the percent error?

1.0%
10%
11%
correctIndex eccentricity

Show answer and explanation

10%

The difference is

1.0\ \mathrm{m/s}

. Dividing by the

10.0\ \mathrm{m/s}

reference and multiplying by 100%

gives

10%.

Key terms

Evidence: Observations or measurements used to assess a physics claim.
System: The object or group of objects being studied.
Reference frame: The viewpoint or coordinate system used to describe position and motion.
Uncertainty: An estimated range that describes the limits of a measurement.
Random error: Unpredictable variation that can make repeated measurements differ.
Systematic error: A consistent shift in measurements, often linked to a bias in a method or instrument.
Percent error: The magnitude of the difference from a reference, expressed as a percentage of that reference.

Continue through SPH4U

View the complete SPH4U Ontario Grade 12 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation A1.8. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

A1.8 · Analyse evidence, solve quantitative problems, and evaluate error

Ontario Grade 12 Physics

A practical framework for analysing physics results at the Grade 12 level

What you will learn

1. Set up the system and read the evidence

2. Choose a relationship and solve with units

3. Describe and evaluate error

4. Make the evidence-based conclusion

Finding average velocity from position evidence

Using net force evidence

Evaluating error against a reference value

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH4U

About this lesson and its review