IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The aim is now complete and clearly connected to the comparison being made. The student explains that the investigation will compare a solar-sail spacecraft with a conventional rocket, which gives the reader a strong roadmap and fixes the abruptness noted in the earlier draft.

The variables table is a strong communication choice because it gives the reader a shared reference for the notation before the derivations begin. This helps prevent ambiguity and is especially useful in a long modelling exploration.

Some variable definitions are imprecise, such as describing \vec{v}_g as “gravitational drag from the Sun.” Drag is not the right term for this context; the student should use a more accurate label, such as gravitational force or gravitational acceleration, to keep the language physically correct.

The rocket equation is introduced appropriately, but the notation should distinguish clearly between the standard gravity term and the constant actually used later. The student should define the symbol consistently and use the accepted value for $g_0$ to avoid undermining the calculation’s precision.

The rearrangement is mathematically on the right track, but the substitution step should be written with clearer dimensional control. In particular, the student should show how the exponent is formed from dimensionless quantities before giving the numerical multiplier, so the reader can verify each stage.

The equation is a promising start, but the student needs to be careful about the difference between pressure and force. If $P_0$ is pressure, the sail area must be included explicitly to obtain force before applying Newton’s second law; otherwise the units do not balance.

The chain-rule step is not expressed correctly here. The student should show the full relationship between $\frac{dv}{dt}$, $\frac{dv}{dr}$, and $\frac{dr}{dt}$, because the present line compresses the derivation in a way that makes the mathematical logic difficult to check.

Including the graph alongside the derivation is effective communication: it lets the reader see the inverse-square trend visually while the algebra is being developed. This is a good example of using multiple representations to support the same idea.

The inverse-square graph is relevant and useful, but a more detailed caption would make it easier to interpret. The student should state the parameter values used and explain what feature of the graph is intended to support the argument about sail performance.

This line mixes force language with pressure notation, which weakens mathematical precision. The student should keep the symbols consistent: if the term is pressure, it must be converted to force using area; if it is force, it should be labeled as such throughout the derivation.

The final numerical time estimate is clearly stated and easy to identify. After a long derivation, this direct presentation of the result helps the reader see the outcome of the model immediately.

The student’s personal motivation is explicit and authentic. Linking the topic to an interest in space exploration and future technology gives the exploration a clear personal direction rather than making it feel generic.

Choosing Falcon Heavy as a benchmark shows independent decision-making about what reference point is most meaningful for the comparison. This makes the costing section feel deliberately selected rather than copied from a template.

To move this from a promising idea to a more distinctive personal contribution, the student should quantify the spiral trajectory more fully. For example, estimating how changing sail orientation or area-to-mass ratio affects travel time would make the initiative much more original.

The proposed spiral/coast-phase idea is a strong sign of initiative because the student is not just comparing two standard transfer methods; they are attempting to adapt the mission profile to the sail’s strengths. That kind of modelling choice adds ownership to the investigation.

Which single modelling change would most improve the investigation if the student had more time: a 3D orbital model, a variable sail orientation model, or a detailed cost sensitivity study? Thinking about that trade-off would sharpen the personal direction of future work.

Including code in the appendix shows initiative and makes the computational approach transparent. It also demonstrates that the numerical result was produced through a repeatable method rather than asserted without support.

The derivation of the Hohmann transfer starts from a standard orbital relationship and moves clearly toward a usable transfer-time expression. This is a strong example of relevant mathematics being applied in a coherent modelling context.

Substituting the constants into the Kepler-based formula shows good command of the mathematical process. The student has kept the derivation tied to the physical quantities rather than jumping straight to a calculator result.

The step toward the delta-v calculation is mathematically relevant, but the notation and terminology should remain consistent with the transfer being analysed. The student should ensure the same orbital model and naming conventions are used throughout so the reader can follow the logic without ambiguity.

The burn calculation is broken into a clear sequence from orbital speed to velocity change. That stepwise structure makes the use of the vis-viva relationship and the resulting $\Delta v$ much easier to verify.

The numerical use of $348\text{s}$ and $9.18{\text{m/s}}^2$ is a reliability issue because the constants are not standard and are not justified in the working shown. The student should either cite the source of the values or replace them with the accepted constants used in the rest of the model.

The solar-sail model introduces calculus in a way that is mathematically appropriate for HL. Using a differential equation to represent changing acceleration shows that the student is engaging with the problem at a suitably advanced level.

The integration setup is relevant, but the expression still contains a conceptual mismatch between the variable being integrated and the physical quantity being modelled. The student should verify that the differential equation matches the intended force balance before integrating, otherwise the resulting formula may not represent the real system.

Moving from an intractable analytic expression to numerical calculation is a sensible mathematical choice at this level. The student has chosen a method that fits the complexity of the model rather than forcing an unrealistic closed-form solution.

The reported times are used meaningfully in the conclusion rather than left as isolated figures. That helps show that the mathematics is serving the research question, not just producing calculations for their own sake.

The appendix suggests the code is helping generate the results, but the main body should still show enough mathematical working to make the method reproducible on its own. The student should avoid relying on the appendix to carry the core reasoning of the exploration.