Measurement, Vectors & Kinematics

Mastery Examination (Level 5)

Time limit: 90 minutes
Total marks: 60
Instructions: Answer all three questions. Show full derivations, justify sign conventions, and state where you use dimensional or vector reasoning. Calculator permitted; take $g = 9.8\ \text{m/s}^2$.

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Question 1 — Dimensional Modelling & Error Propagation (20 marks)

A student models the terminal speed of a small sphere of radius $r$ falling through a viscous fluid. She hypothesises that the drag force has the form $F_d = k\,\eta^{a}\, r^{b}\, v^{c},$ where $\eta$ is the fluid's dynamic viscosity (SI units $\text{Pa·s} = \text{kg·m}^{-1}\text{s}^{-1}$), $v$ is the speed, $k$ is a dimensionless constant.

(a) Using dimensional analysis, determine the exponents $a$, $b$, $c$. State the resulting form of $F_d$. (6)

(b) At terminal velocity the drag balances the net downward force $(\rho_s - \rho_f)\tfrac{4}{3}\pi r^3 g$ (sphere density $\rho_s$, fluid density $\rho_f$). Assuming $k=6\pi$, derive an explicit expression for the terminal speed $v_T$. (5)

(c) In an experiment the following are measured with uncertainties: $r = (2.00 \pm 0.05)\ \text{mm}$, $\eta = (1.00 \pm 0.02)\ \text{Pa·s}$, and the density difference $(\rho_s-\rho_f) = (1200 \pm 30)\ \text{kg·m}^{-3}$. Compute $v_T$ and its percentage uncertainty (propagate errors). Give the final answer to the correct number of significant figures. (6)

(d) State one systematic and one random error that could arise in this experiment, and explain how each would affect the measured $v_T$. (3)

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Question 2 — Vector Algebra, Torque & Work (18 marks)

Three points are given: $A(1,\,0,\,2)$, $B(3,\,2,\,-1)$, $C(0,\,4,\,1)$ (metres).

(a) Find the unit vector along $\vec{AB}$ and the angle $\angle BAC$ using the dot product. (6)

(b) A force $\vec{F} = (4\hat{i} - 3\hat{j} + 2\hat{k})\ \text{N}$ acts at $B$. Taking $A$ as the pivot, compute the torque $\vec{\tau} = \vec{r}\times\vec{F}$ (with $\vec{r}=\vec{AB}$) and its magnitude. Confirm your torque direction is consistent with the right-hand rule. (6)

(c) The same force $\vec F$ moves a particle along displacement $\vec{AC}$. Compute the work done, and hence the component of $\vec F$ along $\vec{AC}$. (3)

(d) Compute the area of triangle $ABC$ using a cross product, and prove in general that $\tfrac12|\vec{AB}\times\vec{AC}|$ equals the triangle's area. (3)

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Question 3 — Kinematics: Calculus, Projectiles & Relative Motion (22 marks)

(a) Starting from the definitions $v=\dfrac{dx}{dt}$ and constant $a=\dfrac{dv}{dt}$, derive by integration the two SUVAT relations $v = u + at$ and $x = ut + \tfrac12 at^2$. Then eliminate $t$ to obtain $v^2 = u^2 + 2ax$. (6)

(b) A projectile is launched from ground level with speed $u$ at angle $\theta$ above the horizontal. From the independent horizontal/vertical equations, derive expressions for the time of flight $T$, maximum height $H$, and range $R$. Prove that the range is maximised at $\theta = 45^\circ$. (6)

(c) A ball is thrown at $u = 25\ \text{m/s}$, $\theta = 40^\circ$. Compute $T$, $H$, $R$, and the speed and direction of the velocity vector at $t = 1.5\ \text{s}$. (6)

(d) A river flows east at $3\ \text{m/s}$. A boat can move at $5\ \text{m/s}$ relative to the water. The pilot wants to cross a $120\ \text{m}$ wide river (banks run east–west… i.e. river flows east, crossing is north) and land directly opposite the start. Determine the heading the pilot must steer (angle west of north), the boat's ground speed, and the crossing time. Use a Galilean velocity-addition argument. (4)

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