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IBDP Physics- A.5 Galilean and special relativity- IB Style Questions For HL Paper 2 -FA 2025

IBDP Physics HL Paper 2 -FA 2025- All Chapters

Question

The space–time diagram below shows the coordinate axes (\(x\), \(ct\)) for an inertial reference frame \(S\). Two events, labelled P and Q, are indicated on the diagram.
(a) (i) Determine the space–time interval \((\Delta s)^{2}\) between events P and Q.
(ii) The quantity \((\Delta s)^{2}\) is invariant. Outline what is meant by an invariant quantity.
Events P and Q are simultaneous in another inertial reference frame \(S’\). The clocks in frames \(S\) and \(S’\) are synchronized when \(x = x’ = 0\).
(b) (i) On the same diagram, sketch the coordinate axes (\(x’\), \(ct’\)) for the frame \(S’\).
(ii) Find the speed of frame \(S’\) relative to frame \(S\).

Most-appropriate topic codes (IB Physics):

• Topic A.5: Galilean and special relativity (spacetime diagrams, invariant interval) — parts (a), (b)
▶️ Answer/Explanation
Detailed solution

(a)
(i) From the diagram, event P has coordinates \((x=2, ct=2)\) and event Q has coordinates \((x=5, ct=3)\).
\(\Delta x = 5 – 2 = 3\,m\).
\(c\Delta t = 3 – 2 = 1\,m\).
\[ (\Delta s)^2 = (c\Delta t)^2 – (\Delta x)^2 = 1^2 – 3^2 = -8\,m^2. \] (ii) An invariant quantity is one that has the same numerical value in all inertial reference frames.

(b)
(i) Because P and Q are simultaneous in frame \(S’\), the straight line joining P and Q represents the \(x’\)-axis (a line of constant \(ct’\)). This line passes through the origin and is parallel to PQ. The \(ct’\)-axis corresponds to \(x’=0\) and is symmetric to the \(x’\)-axis about the light cone. The slope of the \(x’\)-axis is \(1/3\), so the slope of the \(ct’\)-axis is \(3\).


(ii) On a \((x, ct)\) diagram, the slope of the \(x’\)-axis gives \(\beta = v/c\).
\[ \text{slope} = \frac{\Delta ct}{\Delta x} = \frac{1}{3} \Rightarrow \frac{v}{c} = \frac{1}{3}. \] Hence, \[ v = \frac{c}{3} \approx 1.0 \times 10^{8}\,m\,s^{-1}. \]

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