Understand · explore · practise
Turn a real motion question into assumptions, variables and equations, then test the prediction and refine the model.
Before you startSubstitution, rearranging equations and basic units.
01 / Begin with a question you can answer
A calculation is only one part of modelling.
Define the question, choose assumptions, represent the system mathematically, solve, interpret, compare with evidence and revise when needed. A useful simple model need not describe every detail.
A cart travels along a straight 12-metre track. Model its position as x=2t, with x in metres and elapsed time t in seconds, until it reaches the end. This assumes constant speed 2 m/s and a start at x=0.
02 / Define the system and the target
For a cart crossing a 12 m track, distinguish a question about travel time from one about maximum safe load.
The quantities and assumptions needed differ.
Travel time asks how long the motion takes and needs a motion model. Maximum safe load concerns forces and material or stability limits; knowing the speed alone is insufficient.
Define x and t for a straight-track position model, including an origin.
Units and reference points are part of the definition.
Let t be elapsed time in seconds since the cart starts, and x be position in metres measured from the start towards the far end. Then x=0 at t=0.
03 / Say which features you idealise
Give a situation in which constant speed is a useful approximation and one in which it is poor.
Think about startup and steady motion.
It may approximate the middle of a steadily driven journey. It poorly describes a substantial startup or braking phase where speed changes.
If the cart is represented by a point, does that automatically mean air resistance is zero?
Geometry and forces are separate modelling choices.
No. A particle model neglects the extent or rotation relevant to the problem, but a force such as drag can still be included. Negligible air resistance is a separate assumption.
04 / Translate assumptions into a relation
x=2t for 0≤t≤6
The factor 2 has units m/s. This relation begins at x=0.
12=2t, so t=6 s
Solve for the event that ends this modelled journey.
At t=3 s, x=6 m
Interpret the value as a position from the chosen origin.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
If the cart starts at x=1 m with the same constant speed, write its position equation.
Include the initial position.
x=1+2t, with x in metres and t in seconds. If the endpoint remains x=12 m, it reaches it after (12−1)/2=5.5 s.
Why would x=2+t be invalid if 2 is a length and t is a time without a conversion factor?
Only like dimensions can be added.
It would add metres to seconds. A velocity factor multiplying time is needed to produce a length; coefficients in physical equations carry units.
05 / Put the answer back in the physical setting
Substituting t=8 gives x=16. Does that predict a real position on the 12 m track?
The model was restricted to the journey before the endpoint.
No. It is an extrapolation of the equation beyond its stated domain. The cart may stop, turn or leave the track; a new model is needed.
What does t=−1 represent algebraically, and why is it excluded here?
Time is measured from the start of this experiment.
The equation gives x=−2, but t=−1 is one second before the defined start. The model was only proposed for elapsed times from 0 to 6 s.
06 / Compare predictions with observations
In a constructed trial, the measured crossing time is 6.4 s. The model predicted 6.0 s. Treat these as stated values for the calculation; actual measurements also have uncertainty.
Find the absolute time discrepancy and its percentage relative to the measured time.
Use 6.4 as the requested reference.
Absolute discrepancy is 0.4 s. Relative to 6.4 s it is (0.4/6.4)×100=6.25%. A percentage relative to the prediction would use a different denominator.
Does a nonzero discrepancy automatically make the model useless?
The required accuracy depends on the purpose.
No. Compare the discrepancy with measurement uncertainty and the accuracy needed. A rough scheduling estimate may tolerate an error that precision control would not.
07 / Improve the assumption that matters
Suggest a refinement if the cart takes time to accelerate from rest.
One constant speed may not cover the whole journey.
Use a separate acceleration phase followed by steady motion, with parameters based on measurements. Match position and time consistently between the phases.
Why test a revised model on another trial rather than only the data used to fit it?
Fitting data and predicting new data are different.
A model can match its fitting data yet predict poorly elsewhere. A new trial provides a check of how well the revised assumptions generalise.
What constant speed would reproduce a 12 m crossing in 6.4 s? Does this prove the actual speed was constant?
Calculate distance divided by time.
12/6.4=1.875 m/s. It is the journey average speed; changing speeds can produce the same total time.
State three things that should accompany a numerical model prediction.
Make the conditions and interpretation visible.
For example: units and reference direction, the assumptions used, and the valid time or position range. Also state relevant uncertainty or limitations.
08 / Use a model as a claim you can test
Question → assumptions and variables → equation → solution → interpretation → comparison with evidence → refinement. The arrows are a workflow, not a guarantee of truth. A clear domain and a meaningful check are as important as correct arithmetic.
Section 1 of 8 · Begin with a question you can answer