Bertozzi's Experiment: Evidence for Relativistic Kinetic Energy Revision Notes | A Level AQA Physics
Bertozzi's 1964 experiment accelerated electrons to known kinetic energies and timed them over 8.4 m to find their speed. Free notes and 13 questions answered.
Bertozzi's Experiment: Evidence for Relativistic Kinetic Energy is part of Special relativity (A-level only) in A Level AQA Physics.
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What this lesson covers
Bertozzi's 1964 experiment accelerated electrons to known kinetic energies and timed them over 8.4 m to find their speed. Free notes and 13 questions answered.
Describe the key features of Bertozzi's experiment and the measurements it produced
Explain how Bertozzi's results provide direct evidence that kinetic energy increases without limit as speed approaches c
Compare the predictions of Newtonian mechanics with the relativistic relationship between kinetic energy and speed
Evaluate Bertozzi's experiment as evidence for special relativity, identifying its strengths and limitations
Key ideas to keep in view
Use this lesson to stay inside Special relativity (A-level only) while connecting the detail back to the wider Turning points in physics (A-level only) topic.
Relativistic kinetic energy: The kinetic energy of a particle moving at speeds comparable to c, given by $E_k = (\gamma - 1)mc^2$, where $\gamma$ is the Lorentz factor. It diverges to infinity as $v \to c$, unlike the Newtonian expression $\frac{1}{2}mv^2$.
Lorentz factor ($\gamma$): A dimensionless factor defined as $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$, which quantifies how much relativistic effects deviate from Newtonian predictions. It equals 1 at rest and increases without bound as $v \to c$.
Speed limit of c: The speed of light in a vacuum, $c \approx 3.0 \times 10^8$ m/s, which acts as an absolute upper limit on the speed of any particle with mass. No finite amount of energy can accelerate a massive particle to c.
Newtonian kinetic energy: The classical expression $E_k = \frac{1}{2}mv^2$, which predicts that speed increases without limit as energy is added. Valid only when $v \ll c$; it significantly underestimates the energy required at relativistic speeds.
Revision notes
Read the core explanations from Bertozzi's Experiment: Evidence for Relativistic Kinetic Energy before testing yourself from memory.
1. The Problem: Can You Keep Accelerating Forever?
Imagine you are pushing a particle harder and harder, pouring in more and more energy. Newtonian mechanics says the particle should keep speeding up indefinitely — double the kinetic energy, and you expect a predictable increase in speed. But what if, at very high speeds, something fundamentally different happens?
That is precisely the question William Bertozzi set out to answer experimentally in 1964. His experiment is one of the most direct pieces of evidence that Newtonian mechanics breaks down at relativistic speeds, and that Einstein's special relativity correctly describes the relationship between kinetic energy and speed.
To appreciate what Bertozzi found, you first need to see what the two theories actually predict — and where they diverge.
2. Two Predictions: Newton vs Einstein
In Newtonian mechanics, kinetic energy is $E_k = \frac{1}{2}mv^2$. Rearranging gives $v = \sqrt{\frac{2E_k}{m}}$, which means speed grows without bound as you add energy — there is no upper limit on $v$.
Special relativity tells a very different story. The relativistic kinetic energy is $E_k = (\gamma - 1)mc^2$, where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$. As $v$ approaches $c$, $\gamma$ increases without limit, meaning you need an infinite amount of energy to reach $c$. The speed of the particle asymptotically approaches $c$ but never reaches it.
The critical difference is this: Newton predicts a parabolic relationship between $E_k$ and $v$ with no ceiling on speed, while relativity predicts that the curve flattens as $v \to c$, with $c$ acting as an asymptote. Bertozzi's experiment was designed to measure which prediction matches reality.
Notice how the two curves agree at low speeds but diverge dramatically as energy increases. At very high kinetic energies, the relativistic curve flattens towards $c$, while the Newtonian curve continues rising. Bertozzi's experiment measured real data points on this graph.
3. Bertozzi's Experiment: The Method
Bertozzi accelerated electrons through a linear accelerator (linac), giving them a precisely known amount of kinetic energy by controlling the accelerating voltage. He then measured the actual speed of those electrons independently, using a time-of-flight method over a known distance.
The time-of-flight measurement worked as follows: electrons travelled a fixed distance of 8.4 m from the end of the accelerator to an aluminium disc target. By measuring the time taken for the electrons to travel this distance, Bertozzi could calculate their speed directly as $v = \frac{d}{t}$. The kinetic energy was determined separately from the accelerating conditions and confirmed calorimetrically — the electrons were stopped in the aluminium disc, and the heat produced was measured to verify the energy delivered.
This two-pronged approach — independent measurement of both energy and speed — is what makes the experiment so compelling. Neither measurement relied on assuming a particular theory of mechanics.
Bertozzi repeated the measurement at several different accelerating energies, building up a set of data points: measured speed plotted against known kinetic energy. He could then overlay these points on both the Newtonian and relativistic theoretical curves to see which theory the data supported.
4. The Results: What Bertozzi Found
The results were striking. At low kinetic energies, the measured speeds agreed with both Newtonian and relativistic predictions — the two theories are nearly identical when $v \ll c$. However, as the kinetic energy was increased substantially, the measured speeds fell well below the Newtonian prediction and instead tracked the relativistic curve closely.
Crucially, even as the kinetic energy was increased to very large values, the measured speed of the electrons never reached $c$. The electrons approached $c$ asymptotically, exactly as special relativity predicts. The Newtonian prediction — that the electrons should have exceeded $c$ at those energy levels — was clearly falsified.
The calorimetric check confirmed that the energy delivered to the aluminium target matched the kinetic energy calculated from the accelerating conditions, ruling out the possibility that energy was being lost elsewhere. The experiment was internally consistent and the conclusion was unambiguous: the relativistic expression for kinetic energy is correct.
5. Calculating the Lorentz Factor and Relativistic Kinetic Energy
To see why the Newtonian prediction fails so dramatically at high speeds, it helps to calculate $\gamma$ and compare the two energy expressions for a concrete case. The following worked example uses values representative of the kind of speeds Bertozzi's electrons reached.
Worked example: Comparing Newtonian and Relativistic Kinetic Energy at $v = 0.98c$
An electron (rest mass $m_e = 9.11 \times 10^{-31}$ kg) travels at $v = 0.98c$, where $c = 3.0 \times 10^8$ m/s. Calculate (a) the Lorentz factor $\gamma$, (b) the relativistic kinetic energy $E_k$, and (c) the Newtonian kinetic energy. Compare the two results.
$\frac{v^2}{c^2} = (0.98)^2 = 0.98 \times 0.98 = 0.9604$ — Always square the fraction of c first. Note: $0.98^2 = 0.9604$, not 0.98 — expand the multiplication explicitly.
$1 - 0.9604 = 0.0396$ — This small number is what makes $\gamma$ large at high speeds.
$\gamma = \frac{1}{\sqrt{0.0396}} = \frac{1}{0.199} \approx 5.03$ — $\sqrt{0.0396} \approx 0.199$. The Lorentz factor is about 5 — meaning the relativistic energy is roughly 5 times the rest energy contribution.
$E_k^{\text{Newton}} = \frac{1}{2}m_e v^2 = \frac{1}{2} \times 9.11 \times 10^{-31} \times (0.98 \times 3.0 \times 10^8)^2$\n$v = 0.98 \times 3.0 \times 10^8 = 2.94 \times 10^8$ m/s\n$v^2 = 2.94 \times 2.94 \times 10^{16} = 8.6436 \times 10^{16}$ m$^2$s$^{-2}$\n$E_k^{\text{Newton}} = 0.5 \times 9.11 \times 10^{-31} \times 8.6436 \times 10^{16} \approx 3.94 \times 10^{-14}$ J — The Newtonian result is about 8 times smaller than the relativistic result. At $v = 0.98c$, Newtonian mechanics drastically underestimates the kinetic energy — which means it would also drastically overestimate how fast a particle should be moving for a given energy input.
$\frac{E_k^{\text{rel}}}{E_k^{\text{Newton}}} \approx \frac{3.30 \times 10^{-13}}{3.94 \times 10^{-14}} \approx 8.4$ — The relativistic kinetic energy is approximately 8.4 times larger than the Newtonian value at this speed. Bertozzi's experiment confirmed that the actual measured speeds matched the relativistic prediction, not the Newtonian one.
Answer: $\gamma \approx 5.03$; relativistic $E_k \approx 3.30 \times 10^{-13}$ J; Newtonian $E_k \approx 3.94 \times 10^{-14}$ J. The relativistic value is approximately 8.4 times larger, confirming that Newtonian mechanics fails severely at $v = 0.98c$.
This calculation makes the physical meaning of Bertozzi's results concrete. If you put in enough energy to give an electron a Newtonian speed of, say, $1.5c$, what actually happens is that the electron's speed barely increases beyond $0.99c$ — the extra energy goes into increasing $\gamma$, not into increasing $v$.
6. Why This Experiment Matters: Evaluating the Evidence
Bertozzi's experiment is significant not because it was the first hint of relativistic behaviour, but because it provided a direct, controlled, quantitative test. The speed was measured independently of the energy, so the comparison between theory and experiment was not circular.
The calorimetric confirmation of the energy values added a second independent check, strengthening the reliability of the results. The experiment covered a range of energies, producing multiple data points that collectively traced the relativistic curve — this is more convincing than a single measurement.
A limitation worth noting is that the experiment used electrons, so it directly tests the relativistic behaviour of a specific particle. However, the same asymptotic behaviour has since been confirmed for protons and other particles in accelerators such as the Large Hadron Collider, where particles routinely reach speeds of $0.9999c$ and beyond without ever reaching $c$.
7. Exam Technique: Answering Questions on Bertozzi's Experiment
AQA questions on this experiment tend to focus on three areas: describing the method, explaining what the results show, and evaluating the experiment as evidence. Each requires a different approach.
For a describe question about the method, work through the sequence: electrons are accelerated to a known kinetic energy, travel a fixed distance, and the transit time is measured to give speed via $v = d/t$. The calorimetric check confirms the energy. Do not explain why — just state what happens in order.
For an explain question about the results, you need cause and effect. A strong answer states that as kinetic energy increases, the measured speed approaches $c$ asymptotically rather than increasing without limit, because the relativistic kinetic energy expression $E_k = (\gamma - 1)mc^2$ requires infinite energy to reach $c$. Contrast this explicitly with the Newtonian prediction.
For an evaluate question, structure your response as: strength — independent measurement of speed and energy; strength — multiple data points; limitation — only electrons tested directly; conclusion — results are consistent with special relativity and inconsistent with Newtonian mechanics, so the experiment provides strong evidence for the relativistic expression.
Bertozzi's experiment directly addresses the AQA specification requirement for evidence of the variation of kinetic energy with speed: it provides a controlled, quantitative demonstration that measured electron speeds match the relativistic prediction and not the Newtonian one, confirming that kinetic energy increases without limit as speed approaches c.
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Published by Revise Right. Use the official specification reference to check the requirements for your course.
Bertozzi's 1964 experiment accelerated electrons in a linear accelerator to known kinetic energies and measured their speed by timing them over 8.4 m to an aluminium disc. It showed that speed approaches c but never reaches it as kinetic energy increases, confirming the relativistic kinetic energy equation.
How did Bertozzi measure the speed of the electrons?
Bertozzi measured electron speed by time of flight: electrons travelled a fixed distance of 8.4 m from the end of the accelerator to an aluminium disc target, and v = d/t. The kinetic energy was confirmed calorimetrically by measuring the heat produced when the electrons stopped in the disc.
What did Bertozzi's results show?
Bertozzi's results showed that at low kinetic energies measured speeds matched both theories, but at high energies they fell well short of the Newtonian prediction and tracked the relativistic curve. Even at very large energies the electrons never reached c, so the Newtonian prediction was clearly falsified.
How do Newtonian and relativistic kinetic energy compare at 0.98c?
Relativistic kinetic energy at 0.98c is about 8.4 times the Newtonian value. For an electron, γ ≈ 5.03 gives (γ − 1) × rest mass × c² ≈ 3.30 × 10⁻¹³ J, while the Newtonian ½mv² gives only about 3.94 × 10⁻¹⁴ J.
What are the strengths and limitations of Bertozzi's experiment?
Bertozzi's experiment measured speed and energy independently, so the comparison was not circular, and it produced multiple data points across a range of energies. Its limitation is that it directly tested only electrons, although the same behaviour has since been confirmed for protons in the Large Hadron Collider.
How did Bertozzi measure the kinetic energy of the electrons?
Bertozzi found the kinetic energy of the electrons from the accelerating conditions in a linear accelerator and confirmed it calorimetrically. The electrons were stopped in an aluminium disc and the heat produced was measured to confirm the energy delivered and ruling out energy being lost elsewhere.
When did Bertozzi do his experiment?
William Bertozzi did his experiment in 1964, accelerating electrons in a linear accelerator and measuring their speed by time of flight. It became one of the most direct pieces of evidence that Newtonian mechanics breaks down at relativistic speeds.
What does Newtonian mechanics predict for speed and kinetic energy?
Newtonian mechanics predicts v = √(2 × kinetic energy ÷ m), from kinetic energy = ½mv², so speed grows without bound as energy is added and there is no upper limit on v. Relativity instead predicts the curve flattens as v approaches c, with c as an asymptote.
Where does the extra energy go as an electron approaches the speed of light?
As an electron approaches the speed of light, extra energy goes into increasing γ rather than increasing v. Enough energy to give an electron a Newtonian speed of 1.5c actually takes its speed barely beyond 0.99c, as Bertozzi's measurements showed.
What does a graph of speed against kinetic energy show?
A graph of speed against kinetic energy shows the Newtonian and relativistic curves agreeing at low energies but diverging as energy increases. The relativistic curve flattens towards c, while the Newtonian curve keeps rising past it; Bertozzi's measured data points followed the relativistic curve.
Has Bertozzi's result been confirmed for other particles?
Bertozzi's result has been confirmed for other particles. The same asymptotic behaviour has been seen for protons in accelerators such as the Large Hadron Collider, where particles routinely reach 0.9999c and beyond without ever reaching c.
Why is Bertozzi's experiment reliable evidence?
Bertozzi's experiment is compelling evidence because speed was measured independently of energy, so the comparison with theory was not circular. The calorimetric check gave a second independent confirmation of the energy values, and multiple data points over a range of energies traced the relativistic curve.
Why did Bertozzi's electrons never reach the speed of light?
Bertozzi's electrons never reached the speed of light because relativistic kinetic energy, (γ − 1)mc², requires infinite energy to reach c. Even at very large kinetic energies the measured speeds approached c asymptotically, and the Newtonian prediction that the electrons should have exceeded c was clearly falsified.