B. Kinematics · Kinematics: Motion in One and Two Dimensions

Investigating Motion: Planning, Measuring and Analysing a Kinematics Inquiry

B2.4 B2.6 A1.1 A1.2 A1.4 A1.5 A1.6 A1.8 A1.10 A1.11About 75 minutes

Learning goals

I can plan a one-dimensional motion inquiry with a testable question, hypothesis, variables and a safe procedure.
I can choose suitable tools (ticker tape, motion sensor, video analysis, simulation) to measure motion.
I can analyse uniform and non-uniform motion data with tables, graphs, slopes and equations.
I can use linearization to find acceleration from a straight-line graph.
I can calculate percent error and percent difference and describe sources of error.

Words to know

independent variable
The variable you deliberately change in an experiment (for example, ramp height).

dependent variable
The variable you measure to see the effect (for example, acceleration).

controlled variables
Everything you keep the same so the test is fair.

ticker-tape timer
A device that marks dots on a paper tape at a fixed rate as an object pulls the tape through it.

linearization
Graphing a changed variable (such as t²) so that a curved relationship becomes a straight line.

percent error
|experimental − accepted| ÷ accepted × 100%.

percent difference
|value 1 − value 2| ÷ (average of the two) × 100%. Used when there is no accepted value.

random error / systematic error
Random errors scatter results unpredictably (reduced by repeating trials). Systematic errors push every result the same way (reduced by fixing the method or tool).

Doing physics, not just reading it

Every equation in this unit was first discovered by someone who measured real motion carefully and looked for patterns. In this lesson you will plan an investigation like a physicist, collect motion data, and turn messy measurements into a reliable result, with an honest statement of how good that result is.

Tools for measuring motion

Ticker-tape timer: marks dots on a tape at a fixed rate, often 60 dots per second. The spacing of the dots shows how far the object moved in each tiny time interval.
Motion sensor: sends out pulses of ultrasound and times the echoes from the object, then plots d–t, v–t and a–t graphs live.
Photogates: a light beam that starts or stops a timer when an object passes through it.
Video analysis: film the motion beside a metre stick and step through frames. Free programs such as Tracker (from the Open Source Physics project) can mark the object in each frame and produce graphs. Phone slow-motion modes work well.
Phone sensors: apps such as phyphox read your phone's built-in accelerometer.
Simulations: interactive programs such as the PhET simulation The Moving Man let you test ideas quickly, though they model ideal situations rather than measuring real ones.

Planning an inquiry

A good plan answers these questions before you touch any equipment:

Question: what exactly are you testing? Make it testable: "How does the height of a ramp affect the acceleration of a cart rolling down it?"
Hypothesis and prediction: "If the ramp is raised, then the acceleration will increase, because a steeper slope makes the cart's speed grow faster." A hypothesis gives a reason; a prediction says what you expect to measure.
Variables: independent (ramp height), dependent (acceleration), controlled (same cart, same ramp surface, same starting point, released from rest, same length of run).
Procedure and tools: numbered steps someone else could repeat. Plan at least three trials for each setting.
Safety: clear the floor of trip hazards, secure the ramp so it cannot slip, put a soft stopper at the end so carts do not fly off, and keep cords tidy (A1.4).
Data table: prepare columns with headings and units before you start.

Worked example

Example 1: Reading a ticker tape

A cart pulls a tape through a 60 Hz ticker timer. Counting every sixth dot (0.10 s apart), the lengths of five successive intervals are 2.0, 3.0, 4.0, 5.0 and 6.0 cm. Is the motion uniform or non-uniform? Find the acceleration.

The intervals get longer by the same amount (1.0 cm) each time, so the cart is speeding up steadily: non-uniform motion with constant acceleration.
Average velocity in each interval = length ÷ 0.10 s: 20, 30, 40, 50 and 60 cm/s.
Each average velocity is the instantaneous velocity at the middle of its interval (true for constant acceleration): at t = 0.05, 0.15, 0.25, 0.35 and 0.45 s.
The velocity rises 10 cm/s every 0.10 s, so a = 10 cm/s ÷ 0.10 s = 100 cm/s² = 1.0 m/s². (A v–t graph of these five points is a straight line with this slope.)

Non-uniform motion with constant acceleration of 1.0 m/s² in the direction of motion.

Example 1. Each gap is 0.10 s of motion. The gaps grow by 1.0 cm each time, the signature of constant acceleration. For uniform motion, all gaps would be equal.

Worked example

Example 2: Linearizing ramp data

A ball is released from rest on a ramp and filmed. Its positions are: t = 0, 0.20, 0.40, 0.60, 0.80, 1.00 s; d = 0, 0.019, 0.071, 0.165, 0.286, 0.452 m. Find its acceleration.

A d–t graph of these points curves upward, so the ball speeds up. A curve is hard to analyse, so linearize.
From rest, Δd = ½aΔt². So d is proportional to t², and a graph of d versus t² should be a straight line through the origin with slope = ½a.
Calculate t²: 0, 0.04, 0.16, 0.36, 0.64, 1.00 s². Plot d against t² and draw the best-fit line through the origin.
The best-fit slope is about 0.451 m/s² (the line passes very close to (1.00, 0.451)). So ½a = 0.451 and a = 0.90 m/s².
The points lie close to the line, which supports the model of constant acceleration.

a ≈ 0.90 m/s² down the ramp.

Plotting d against t² straightens the curve. The slope of the best-fit line is ½a. A straight line through the origin is evidence that the acceleration was constant.

How good is your result?

Percent error compares your result with an accepted value:

percent error = |experimental − accepted| ÷ accepted × 100%

Percent difference compares two of your own results when there is no accepted value (for example, acceleration found from a graph and from an equation):

percent difference = |value 1 − value 2| ÷ (average of the two values) × 100%

Sources of error should be specific. "Human error" is not an answer. Instead:

Random errors scatter your results, for example starting and stopping a stopwatch by hand, or judging which video frame shows the release. Reduce them by repeating trials and averaging.
Systematic errors push every result the same way, for example a metre stick that is not lined up with the ramp, a camera viewing at an angle, or friction you ignored. Reduce them by improving the method.

Worked example

Example 3: Evaluating results

(a) A student measures g as 9.52 m/s². Find the percent error. (b) Another student finds a cart's acceleration as 0.84 m/s² from a v–t graph and 0.90 m/s² from Δd = ½aΔt². Find the percent difference. (c) Suggest one specific source of error and one improvement for (a).

(a) |9.52 − 9.8| ÷ 9.8 × 100% = 0.28 ÷ 9.8 × 100% ≈ 2.9%.
(b) Average = (0.84 + 0.90) ÷ 2 = 0.87. |0.84 − 0.90| ÷ 0.87 × 100% ≈ 6.9%.
(c) A result lower than 9.8 m/s² every time points to a systematic cause, such as air resistance slowing the ball or timing that started a little early (making Δt too large). Improvement: use a denser, more compact ball and find Δt by counting slow-motion video frames rather than using a stopwatch.

(a) 2.9%; (b) 6.9%; (c) see the steps.

Communicating your inquiry (A1.11)

A physics lab report usually has: Purpose/Question, Hypothesis, Materials, Procedure (numbered, past or imperative tense), Observations (tables with units, labelled graphs), Analysis (sample calculations, slopes, percent error), Evaluation (sources of error, improvements) and Conclusion (answer the question, using evidence). A conclusion should say whether the evidence supports the hypothesis, not "prove" it.

Just for fun

Why did the ticker-tape timer never win an argument?

It kept making the same point, sixty times a second.

Hands-on activity

Your own ramp inquiry

You need: a long, stiff board or a large hardcover book set (for a ramp), books to raise one end, a toy car or a ball, masking tape, a metre stick or measuring tape, and a phone with slow-motion video. Grown-up note: set up on the floor, keep the area clear of trip hazards, and put a cushion or towel at the bottom of the ramp.

Question: How does ramp height affect acceleration? Write a hypothesis with a reason.
Lay tape marks every 20 cm along the ramp. Raise one end with books to a measured height (say 5 cm).
Release the car from rest at the top mark. Film it from the side in slow motion, with the tape marks visible.
From the video, find the time at which the car passes each mark (frame number ÷ frames per second). Do 3 trials.
Repeat for a second height (say 10 cm).
For each height: make a table of t, t² and d; graph d versus t²; find a = 2 × slope.
Compare the two accelerations. Calculate the percent difference between two trials at the same height to check consistency. List two specific sources of error and an improvement.

What to look for: a steeper ramp should give a larger acceleration. For a smooth ramp, raising the height from 5 cm to 10 cm (same length) should roughly double the acceleration. Your d versus t² graphs should be close to straight lines through the origin. Trials at the same height usually agree within about 5 to 10% with careful frame counting.

Your turn

In the ramp inquiry, name the independent, dependent and two controlled variables.

A ticker tape (0.10 s between marked dots) shows intervals of 4.0, 4.0, 4.0 and 4.0 cm. Describe the motion and give the velocity.

Why do we graph d against t² for an object starting from rest with constant acceleration?

A d versus t² graph has a slope of 1.6 m/s². Find a.

Find the percent error if you measure g = 10.3 m/s².

Give one random error and one systematic error in timing a dropped ball with a stopwatch.

Rewrite as a testable question: "Do balls roll better on different floors?"

More practice: warm-up

Which variable do you change on purpose?

A 60 Hz ticker timer makes how many dots per second?

How long is the time between dots on a 60 Hz tape?

How do you reduce random error?

Name one tool that can measure motion other than a stopwatch and ruler.

More practice: on your own

Ticker tape intervals (0.10 s each): 9.0, 7.5, 6.0, 4.5 cm. Describe the motion and find the acceleration.

A cart starts from rest and passes 1.20 m after 1.60 s. Find a, and its velocity at that point.

Two methods give a = 2.10 m/s² and 2.30 m/s². Find the percent difference.

A student's d–t graph for a cart is a straight line. She concludes the cart was accelerating. Evaluate her conclusion.

List four safety precautions for a ramp-and-cart experiment in a classroom.

Challenge

A student uses a 60 Hz ticker timer but the power in her school is actually 50 Hz. She assumes 0.10 s for every 6 dots. Is her acceleration too big or too small, and by what factor?

Design (in 4 or 5 steps) an inquiry to test whether a ball rolling down a ramp has constant acceleration.

A ramp experiment always gives an acceleration about 30% lower than the student's prediction, no matter how carefully she times it. Suggest a reason based on physics.

Check your understanding

Quiz

In an experiment testing how ramp height affects acceleration, the dependent variable is:

Equal spacing between dots on a ticker tape shows:

For an object starting from rest with constant acceleration, a graph of d versus t² is:

A measured value of g is 9.31 m/s². The percent error is closest to:

Which is a systematic error?

The best way to reduce random error is to:

For parents and teachers

Big idea: This lesson covers the investigation expectations of the strand: investigating uniform and non-uniform motion (B2.4) and planning and conducting a one-dimensional inquiry (B2.6), along with the scientific investigation skills of strand A (questions, hypotheses, variables, safety, data, graphs, error analysis and communication).

How to help: Help set up the ramp activity safely and film the runs. Ask your teen to explain before starting what they will change, what they will measure and what they will keep the same. Afterward ask, "How sure are you of this result, and what would make it better?"

Common difficulties:

Writing "human error" instead of a specific, physical source of error.
Doing only one trial.
Joining data points dot-to-dot instead of drawing a best-fit line.
Forgetting the factor of 2 when finding a from the slope of a d versus t² graph.