HSC Exam Practice

Sample response. An inertial frame of reference is a reference frame that moves at constant velocity (zero acceleration) in which Newton’s first law holds. A car turning a corner is not an inertial frame because it is accelerating (changing direction), so fictitious forces (e.g. the centrifugal “force” pushing you sideways) appear. A train moving at constant velocity on a straight track is an inertial frame because it does not accelerate; passengers in it observe Newton’s laws to hold without any correction.

Marking notes. 1 mark for correct definition of inertial frame (constant velocity / no acceleration); 1 mark for correctly identifying the turning car as non-inertial with correct reason (changing direction = acceleration); 1 mark for correctly identifying the constant-velocity train as inertial with correct reason.

Sample response. First postulate: The laws of physics are identical in all inertial frames of reference; no experiment can detect absolute motion. Supporting observation: performing identical experiments on a smooth-moving train and on the platform gives the same results — there is no mechanical test for “absolute rest.” Second postulate: The speed of light in vacuum ($c \approx 3.00 \times 10^8$ m/s) is the same for all observers, regardless of the motion of the source or observer. Supporting observation: the null result of the Michelson-Morley experiment showed that the speed of light did not vary with Earth’s direction of motion through the hypothetical aether, consistent with $c$ being invariant.

Marking notes. 1 mark for correct first postulate (laws of physics identical in all inertial frames); 1 mark for valid supporting observation for first postulate; 1 mark for correct second postulate (speed of light constant for all observers); 1 mark for valid supporting observation for second postulate (Michelson-Morley or equivalent).

Sample response. The Michelson-Morley experiment (1887) aimed to detect Earth’s motion through the luminiferous aether by comparing the travel time of light in two perpendicular arms of a precision interferometer. If the aether existed and Earth moved through it at ~30 km/s, the expected fringe shift was 0.4 fringe. The observed shift was less than 0.01 fringe — a null result. This result was significant because it showed that the speed of light did not depend on direction or on Earth’s motion through any medium, undermining the aether hypothesis. It was one of the key observations that motivated Einstein to propose that the speed of light is invariant (second postulate), and thus to discard the concept of absolute motion.

Marking notes. 1 mark for stating the purpose (detect Earth’s motion through the aether by measuring a difference in light speed in perpendicular directions); 1 mark for describing the null result (no detectable fringe shift, speed of light same in all directions); 1 mark for significance (undermined aether, supported constancy of $c$, contributed to Einstein’s second postulate).

Sample response (a). $\gamma = 1/\sqrt{1-(0.98)^2} = 1/\sqrt{0.0396} = 1/0.199 \approx \mathbf{5.03}$ [1].

Sample response (b). $E = \gamma m_e c^2 = 5.03 \times 511\text{ keV} = 2570\text{ keV} \approx \mathbf{2.57\text{ MeV}}$ [1].

Sample response (c). $E_k = (\gamma-1)m_e c^2 = 4.03 \times 511 = 2059\text{ keV} \approx \mathbf{2.06\text{ MeV}}$ [1]. [1 mark for method throughout = consistent use of given equations].

Marking notes. 1 mark for correct $\gamma$ ($\approx 5.03$); 1 mark for correct total energy in MeV; 1 mark for correct $E_k$ in MeV; 1 mark for clear method at each step (formula substituted correctly).

Sample response. (a) Galilean (classical) prediction: laser speed = $c + 0.80c = 1.80c$ relative to Earth. (b) Einstein’s second postulate: Earth observers measure the laser light at exactly $c$. (c) For both observers to measure $c$, our classical notions must be revised: time is not absolute (moving clocks run slow — time dilation), and length is not absolute (moving objects are shorter — length contraction). Space and time adjust together so that the measured speed of light remains $c$ in all inertial frames.

Marking notes. 1 mark for Galilean prediction ($1.80c$) and Einstein’s prediction ($c$); 1 mark for identifying that time dilation and length contraction must occur; 1 mark for coherent explanation of why classical notions of absolute space and time must be abandoned.

Sample response. As $v \to c$, $1 - v^2/c^2 \to 0$, so $\sqrt{1-v^2/c^2} \to 0$ and $\gamma \to \infty$ [1]. The total energy required to accelerate a massive particle is $E = \gamma m_e c^2$; as $\gamma \to \infty$, $E \to \infty$. For example, an electron at $v = 0.9997c$ already requires $E = 40.8 \times 511 \approx 20\,800$ keV. At $v = c$, $\gamma$ would be infinite and the required energy would also be infinite [1]. Since it is physically impossible to supply infinite energy, no massive object can ever reach exactly $c$. Only massless particles (photons) naturally travel at $c$ [1].

Marking notes. 1 mark for showing $\gamma \to \infty$ as $v \to c$ mathematically; 1 mark for linking infinite $\gamma$ to infinite energy via $E = \gamma mc^2$; 1 mark for concluding that infinite energy is impossible and applying a valid numerical example.

Sample response (a). At $v = 0.9997c$, $\gamma = 40.8$. $E = 40.8 \times 511 = 20\,849$ keV $\approx \mathbf{20\,800}$ keV [1]. $E_k = (40.8 - 1) \times 511 = 39.8 \times 511 = 20\,338$ keV $\approx \mathbf{20\,300}$ keV [1].

Sample response (b). As $v/c$ increases, $E_k$ grows much faster than the rest mass energy (511 keV): from 79 keV (15% of rest energy at $v = 0.5c$) to 659 keV (129% at $v = 0.9c$) to 20\,300 keV (3970% at $v = 0.9997c$) [1]. $E_k$ grows without bound as $v \to c$ because $\gamma \to \infty$. This directly supports Einstein’s postulate that massive objects cannot reach $c$: reaching $c$ would require infinite kinetic energy, which is impossible [2].

Sample response (c). Classical: $E_k^{\text{cl}} = \tfrac{1}{2}m_e v^2 = \tfrac{1}{2}m_e(0.5c)^2 = 0.125 m_ec^2 = 0.125 \times 511 = 63.9$ keV [1]. This is close to (but about 19% less than) the relativistic value of 79 keV [1]. At $v = 0.5c$, the classical formula is a reasonable approximation (within ~20%), but it increasingly underestimates energy at higher speeds [1].

Marking notes. Part (a): 1 per correct value (2 marks). Part (b): 1 for describing the trend quantitatively; 1 for linking to $\gamma \to \infty$; 1 for connecting to why $v = c$ is impossible. Part (c): 1 for correct classical calculation ($\approx 64$ keV); 1 for correct comparison to relativistic value; 1 for valid comment on accuracy of approximation.

Sample response. By the late 19th century, Galilean relativity was well-established: the laws of mechanics are the same in all inertial frames, and velocities add simply ($u' = u + v$). However, Maxwell’s electromagnetic wave equations predicted a fixed speed $c = 1/\sqrt{\mu_0 \varepsilon_0}$ without reference to any observer or medium. Physicists interpreted this as light propagating through a hypothetical medium, the luminiferous aether, which defined an absolute rest frame. In 1887, Michelson and Morley attempted to detect Earth’s motion through this aether using a precision interferometer. The expected fringe shift (0.4 fringe at Earth’s orbital speed) was well above the instrument’s sensitivity, yet the observed shift was below 0.01 fringe — a definitive null result. This was a profound anomaly: either no aether existed, or light behaved fundamentally differently from sound. In 1905, Einstein resolved this with two postulates. His first postulate extended Galilean relativity to all of physics: the laws of physics, including electromagnetism, are identical in all inertial frames. His second postulate abandoned the aether entirely: the speed of light in vacuum is the same constant $c$ for all observers regardless of their motion or that of the source. From these two postulates, the Lorentz factor $\gamma = 1/\sqrt{1-v^2/c^2}$ emerges naturally. At low speeds $\gamma \approx 1$ and classical physics is recovered. As $v \to c$, $\gamma \to \infty$. The consequences are: (1) Time dilation: moving clocks run slow by factor $\gamma$ ($\Delta t = \gamma \Delta t_0$); (2) Length contraction: moving objects are shorter along the direction of motion by $1/\gamma$ ($L = L_0/\gamma$); (3) Relativistic energy: $E = \gamma mc^2$, so at rest $E_0 = mc^2$ and the kinetic energy is $E_k = (\gamma - 1)mc^2$. Quantitatively, for an electron at $v = 0.98c$: $\gamma = 5.03$; $E = 5.03 \times 511 = 2570$ keV = 2.57 MeV. This is about 5 times the rest mass energy, a result completely inconsistent with classical mechanics but confirmed by particle accelerator experiments. Einstein’s theory replaced absolute space and time with the invariant speed of light as the cornerstone of physics, and all subsequent tests — muon survival in cosmic rays, GPS corrections, accelerator physics — have confirmed it.

Marking criteria (7 marks). 1 = explains Galilean relativity and why Maxwell’s equations created a problem (fixed $c$ without reference to any frame). 1 = describes the Michelson-Morley experiment and its null result with key quantitative detail. 1 = states both Einstein’s postulates correctly (laws of physics identical in all inertial frames; $c$ constant for all observers). 1 = derives or explains the origin of the Lorentz factor from the postulates ($\gamma = 1/\sqrt{1-v^2/c^2}$, equals 1 at rest, diverges at $v \to c$). 1 = correctly names all three relativistic effects (time dilation, length contraction, relativistic energy) with correct formulas. 1 = provides a correct quantitative example using $E = \gamma mc^2$ or $\Delta t = \gamma \Delta t_0$ with numbers consistent with the lesson. 1 = reaches an integrative evaluative conclusion: special relativity replaced Galilean absolute space/time with the invariant $c$; supported by experiment.