Ticker tape timer: finding speed and acceleration
Key idea: A ticker timer prints dots at equal time intervals, so each gap shows the distance moved in one interval: equal gaps mean constant speed, and longer gaps mean faster motion.
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Guided steps
- Follow the marking process and use the given timer frequency.
- Read the dots in time order and count the gaps.
- Compare long-section reading accuracy with short-section changes in speed.
- Follow the gap count, section time and metre conversion.
- Convert frequency to gap time, then calculate the section’s speed.
- Correct the dot-counting error and calculate the speed.
- Find the section time before working back to timer frequency.
- Compare section-average velocities at their mid-times.
- Calculate the section speeds and use the time between their midpoints.
- Calculate the three speeds and compare their increases.
- Include the skipped section when finding the time between midpoints.
- Read each strip as an average speed for one time interval.
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A ticker tape timer is a clock that writes its own record. It prints dots on a paper tape at equal time intervals while a moving trolley pulls the tape through it. Every gap between two dots took the same time, so the length of a gap shows how far the trolley moved in that time: long gaps mean fast motion and short gaps mean slow motion. From one strip of tape you can find the speed at different times and the acceleration.
How the timer marks the tape
If the figure is wider than its frame, scroll across to read all labels.
An electromagnet makes a metal arm vibrate at a fixed frequency, f. Each time the arm comes down, it presses a dot onto the tape. The time between two dots is one period of the vibration:
A timer that vibrates at 50 Hz makes 50 dots every second, so Δ t = 1/50 s = 0.02 s. Not every timer runs at 50 Hz, so use the frequency or the time interval that the question gives.
To record a motion:
- Fix the timer at one end of the runway and attach the tape to the back of the trolley.
- Thread the tape through the timer’s guides so that it lies flat under the marker.
- Switch the timer on, then release the trolley.
- Stop the trolley before the tape runs out, then switch off.
- Ignore the crowded or overlapping dots at the start. Mark sections that each contain the same number of gaps, and measure their lengths with a rule to the nearest millimetre.
The tape rubs on the timer as it is pulled through, so friction opposes the motion and can change the trolley’s speed or acceleration compared with an otherwise identical run without the tape. Do not assume the recording method leaves the motion unchanged.
Read the motion from the dot spacing
Because the time between dots is fixed, the spacing tells you about the speed:
- equal gaps mean equal distances in equal times: constant speed;
- gaps getting longer mean the trolley moves further in each interval: it is speeding up;
- gaps getting shorter mean it moves less far in each interval: it is slowing down.
Read the dots in the order they were made. The first dot is the one nearest the trolley end of the tape, because that part of the tape was under the marker when the timer started.
Count gaps, not dots. Time passes between one dot and the next, so a section with 6 dots contains 5 gaps and lasts 5Δ t.
Calculate speed from a section
- Choose a section and count the number of gaps, N.
- Measure the length of the section, s, and convert it to metres.
- Find the time for the section: t = NΔ t.
- Calculate the average speed: v = s/t.
A longer section makes the ruler reading error at each end a smaller fraction of the length. But if the speed is changing, a long section averages over more of the motion, so use short sections that each last the same time, with the same number of gaps, to see how the speed changes.
Worked example 1
Speed from a section of tape
Problem
A ticker timer makes a dot every 0.02 s. A section of tape contains 10 gaps and is 18 cm long. Find the average speed over that section.
Show full solution
Find the time for the section
Method
Multiply the number of gaps by the time per gap.Reason
Each gap is one complete time interval between two dots.Working
Answer
t = 10 × 0.02 s = 0.20 sConvert the length
Method
Write 18 cm in metres.Reason
A speed in m s⁻¹ needs metres and seconds.Working
Answer
s = 18 cm = 0.18 mDivide length by time
Working
Answer
v = (0.18 m)/(0.20 s) = 0.90 m s⁻¹
Quick check
Guided practice 1
Start from the timer’s frequency
Problem
A ticker timer runs at f = 40 Hz. A section of tape contains 5 gaps and is 12 cm long. Find the average speed over that section.
Find the interval, the time and the speed
Show the steps Optional working
Show the steps Optional working
Hints
Hint 1
turn the frequency into a time
What does 40 Hz tell you about the number of intervals each second?
Hint 2
scale up to five gaps
Multiply Δ t by 5, then use v = s/t with s in metres.
Show solution step by step
Find the time per gap
Method
Take the reciprocal of the frequency.Reason
Frequency is the number of dots per second, so its reciprocal is the number of seconds per gap.Working
Answer
Δ t = 1/(40 Hz) = 0.025 sFind the section time and length
Method
Multiply by five gaps and convert centimetres to metres.Reason
The time and length must describe the same section, in SI units.Working
Answer
t = 5 × 0.025 s = 0.125 s, s = 0.12 mCalculate the speed
Reason
Average speed is distance divided by time.Working
Answer
v = (0.12 m)/(0.125 s) = 0.96 m/s
Quick check
Spot the mistake 1
Counting dots instead of gaps
Learner response
Diagnose before viewing the correction
Show the steps Optional working
Show solution step by step
Count the intervals
Method
Count the gaps between the dots.Reason
The first dot marks the start of the timing, not the end of an interval.Working
Answer
Six dots contain five gaps, so t = 5 × 0.02 s = 0.10 s.Calculate the correct speed
Method
Convert 15 cm to 0.15 m and divide by the correct time.Reason
The extra interval made the learner’s time too long, and so the speed too small.Working
Answer
v = (0.15 m)/(0.10 s) = 1.5 m s⁻¹
Common mistake
Check your understanding 1 Find the timer’s frequency
A section of tape has 24 gaps and is 1.20 m long. The trolley’s average speed over that section was 2.0 m s⁻¹. Find the frequency of the timer.
Show the steps Optional working
Show answer
Time for the section:
Time per gap:
Frequency:
Estimate acceleration from two sections
To find acceleration you need two velocities and the time between them. Measure the average speed over two sections of tape, each with the same number of gaps, in the direction of motion.
When the acceleration is uniform, the velocity changes steadily, so the average velocity over a section equals the velocity at the middle of that section’s time. Treat each section speed as the velocity at its mid-time, then divide the change in velocity by the time between the two mid-times:
For two sections that follow straight on from each other, the mid-times are exactly one section’s duration apart, not two. If the acceleration is not uniform, the answer is an estimate.
Exam-style question 1
Acceleration from two consecutive sections
Examination question
A ticker timer marks a dot every 0.10 s. Two consecutive five-gap sections measure 0.25 m and 0.35 m. Estimate the trolley’s acceleration. [5 marks]
Show the section time, both speeds and the acceleration
Show the steps Optional working
Show the steps Optional working
Show the steps Optional working
Show solution step by step
Find the time for each section
1 markMethod
Multiply five gaps by the interval.Reason
Both sections have the same number of gaps, so they last the same time.Working
Answer
t section = 5 × 0.10 s = 0.50 sCalculate both section speeds
2 marksMethod
Divide each length by the section time.Reason
Each answer is the average speed over its section. For known motion in one direction with uniform acceleration, this equals the velocity at the section’s mid-time; otherwise that identification is an approximation.Working
Answer
v₁ = (0.25 m)/(0.50 s) = 0.50 m s⁻¹, v₂ = (0.35 m)/(0.50 s) = 0.70 m s⁻¹Divide by the time between mid-times
2 marksMethod
Divide the change in velocity by 0.50 s.Reason
The middle of the first section and the middle of the second are one section’s duration apart.Working
Answer
a ≈ (0.70 m s⁻¹-0.50 m s⁻¹)/(0.50 s) = 0.40 m s⁻²
Self-mark with the mark scheme
Tick each mark point that your answer includes.
Self-mark the section method.
Common mistake
Dividing by 1.0 s, the time across both sections, gives 0.20 m s⁻² and loses the last two marks: the mid-times are 0.50 s apart.
Check your understanding 2 Uniform or non-uniform acceleration?
A ticker timer makes a dot every 0.10 s. A tape is split into three consecutive sections, each with 5 gaps. Their lengths are 0.30 m, 0.42 m and 0.63 m.
- Find the average speed in each section.
- Is the acceleration uniform?
Show the steps Optional working
Show the steps Optional working
Show the steps Optional working
Show answer
Each section lasts 5 × 0.10 s = 0.50 s.
The mid-times are 0.50 s apart, so
The speed increases are 0.24 and 0.42 m/s over equal times, so acceleration is not uniform. The midpoint estimates rise from 0.48 to 0.84 m/s².
Mind stretcher 1 What if a section was skipped?Extension
Two 0.40 s sections of tape give average velocities of 0.50 and 0.90 m s⁻¹ in the same direction. One unused 0.40 s section lies between them. Find the time between the used sections’ midpoints and estimate the acceleration. Assume uniform acceleration within each section.
Show the steps Optional working
Show answer
The mid-times are 0.80 s apart: half of the first section, the whole skipped section and half of the last section.
The used sections run from 0 to 0.40 s and 0.80 to 1.20 s, with mid-times 0.20 and 1.00 s. Dividing by 0.40 s would treat them as adjacent and double the answer. Using each section average as its mid-time velocity is exact for uniform acceleration in one direction and an approximation otherwise.
Turn the tape into a speed–time chart
Cut the tape into strips that each contain the same number of gaps, such as 5. Stick the strips side by side in time order, with their bottom ends level.
Every strip covers the same time, so each strip’s length is proportional to the average speed during that time. The row is a speed–time chart drawn as bars; if the trolley is known to move throughout in one positive direction, the bars can also represent positive average velocities. Equal strip lengths show equal section-average speeds. Equal increases in strip length are consistent with uniform acceleration. Each bar represents a section average, rather than an instantaneous speed.
Common mistakes
- Counting dots instead of gaps: n dots contain n-1 gaps.
- Using the wrong time interval, or using the frequency as if it were a time. Use Δ t = 1/f.
- Forgetting to convert centimetres to metres.
- Finding speed from a single gap, where the ruler reading error is large compared with the length.
- Reading the tape in the wrong order. The first dot is at the trolley end.
- Dividing by the whole time covered by two sections, instead of the time between their mid-times, when finding acceleration.
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