Level 2 — RecallMeasurement, Vectors & Kinematics

Measurement, Vectors & Kinematics

40 marksprintable — key stays hidden on paper

Level 2 (Recall & Standard Problems)

Time: 30 minutes Total Marks: 40

Take $g = 9.8\ \text{m/s}^2$ unless otherwise stated. Show working.

Q1. Define a derived physical quantity and give two examples with their SI units. (3 marks)

Q2. Check whether the equation $v^2 = u^2 + 2as$ is dimensionally consistent, where $v,u$ are velocities, $a$ is acceleration and $s$ is displacement. (3 marks)

Q3. A rectangular plate has length $4.25\ \text{cm}$ and breadth $2.1\ \text{cm}$ . Compute its area, expressing the result to the correct number of significant figures. (3 marks)

Q4. The period of a simple pendulum is measured as $T = (2.00 \pm 0.02)\ \text{s}$ . State the absolute error, the relative error, and the percentage error. (4 marks)

Q5. Given $\vec{A} = 3\hat{i} + 4\hat{j}$ , find: (a) the magnitude of $\vec{A}$ , (b) the unit vector along $\vec{A}$ . (4 marks)

Q6. For $\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{B} = \hat{i} - \hat{j} + 4\hat{k}$ , compute: (a) $\vec{A}\cdot\vec{B}$ , (b) $\vec{A}\times\vec{B}$ . (5 marks)

Q7. A car starts from rest and accelerates uniformly at $2\ \text{m/s}^2$ for $10\ \text{s}$ . Find (a) its final velocity and (b) the distance travelled. (4 marks)

Q8. A ball is thrown vertically upward with an initial speed of $19.6\ \text{m/s}$ . Using $g = 9.8\ \text{m/s}^2$ , find the maximum height reached and the total time of flight. (5 marks)

Q9. A projectile is launched with speed $20\ \text{m/s}$ at $30^\circ$ above the horizontal. Find its time of flight, maximum height, and horizontal range. (6 marks)

Q10. A boat can move at $5\ \text{m/s}$ in still water. It heads straight across a river flowing at $3\ \text{m/s}$ . Find the resultant speed and the magnitude of the boat's velocity relative to the ground. (3 marks)

End of paper

Answer keyMark scheme & solutions

Q1. (3 marks) A derived quantity is one expressed as a combination of the seven fundamental (base) quantities. (1) Examples: velocity — $\text{m/s}$ ; force — $\text{N} = \text{kg·m/s}^2$ (any two valid). (1 + 1)

Q2. (3 marks) $[v^2] = (\text{m/s})^2 = \text{m}^2\text{s}^{-2}$ . (1) $[2as] = (\text{m/s}^2)(\text{m}) = \text{m}^2\text{s}^{-2}$ , and $[u^2]=\text{m}^2\text{s}^{-2}$ . (1) All terms have identical dimensions $\Rightarrow$ equation is dimensionally consistent. (1)

Q3. (3 marks) Area $= 4.25 \times 2.1 = 8.925\ \text{cm}^2$ . (1) Least number of sig figs among factors = 2 (from $2.1$ ). (1) Rounded: $A = 8.9\ \text{cm}^2$ . (1)

Q4. (4 marks) Absolute error $= 0.02\ \text{s}$ . (1) Relative error $= \dfrac{0.02}{2.00} = 0.01$ . (1 + 1) Percentage error $= 0.01 \times 100 = 1\%$ . (1)

Q5. (4 marks) (a) $|\vec A| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ . (2) (b) $\hat A = \dfrac{\vec A}{|\vec A|} = \dfrac{3\hat i + 4\hat j}{5} = 0.6\,\hat i + 0.8\,\hat j$ . (2)

Q6. (5 marks) (a) $\vec A\cdot\vec B = (2)(1)+(3)(-1)+(-1)(4) = 2-3-4 = -5$ . (2) (b) $\vec A\times\vec B = \begin{vmatrix}\hat i & \hat j & \hat k\\ 2 & 3 & -1\\ 1 & -1 & 4\end{vmatrix}$ $= \hat i(3\cdot4-(-1)(-1)) - \hat j(2\cdot4-(-1)(1)) + \hat k(2(-1)-3\cdot1)$ $= \hat i(12-1) - \hat j(8+1) + \hat k(-2-3) = 11\hat i - 9\hat j - 5\hat k$ . (3)

Q7. (4 marks) (a) $v = u + at = 0 + 2\times10 = 20\ \text{m/s}$ . (2) (b) $s = ut + \tfrac12 at^2 = 0 + \tfrac12(2)(100) = 100\ \text{m}$ . (2)

Q8. (5 marks) Max height: $v^2 = u^2 - 2gH \Rightarrow 0 = 19.6^2 - 2(9.8)H$ (1) $H = \dfrac{384.16}{19.6} = 19.6\ \text{m}$ . (2) Time of flight: $t = \dfrac{2u}{g} = \dfrac{2(19.6)}{9.8} = 4\ \text{s}$ . (2)

Q9. (6 marks) $u_x = 20\cos30^\circ = 17.32\ \text{m/s}$ , $u_y = 20\sin30^\circ = 10\ \text{m/s}$ . (1) Time of flight $t = \dfrac{2u_y}{g} = \dfrac{20}{9.8} = 2.04\ \text{s}$ . (2) Max height $H = \dfrac{u_y^2}{2g} = \dfrac{100}{19.6} = 5.10\ \text{m}$ . (2) Range $R = \dfrac{u^2\sin2\theta}{g} = \dfrac{400\sin60^\circ}{9.8} = \dfrac{346.4}{9.8} = 35.3\ \text{m}$ . (1)

Q10. (3 marks) Boat velocity across = $5\ \text{m/s}$ , current along stream = $3\ \text{m/s}$ , perpendicular. (1) Resultant $= \sqrt{5^2 + 3^2} = \sqrt{34} = 5.83\ \text{m/s}$ . (2)

[
  {"claim":"Q5 magnitude of A is 5","code":"result = (sqrt(3**2+4**2)==5)"},
  {"claim":"Q6 dot product = -5","code":"A=Matrix([2,3,-1]); B=Matrix([1,-1,4]); result = (A.dot(B)==-5)"},
  {"claim":"Q6 cross product = 11i-9j-5k","code":"A=Matrix([2,3,-1]); B=Matrix([1,-1,4]); result = (A.cross(B)==Matrix([11,-9,-5]))"},
  {"claim":"Q8 max height = 19.6 m","code":"u=Rational(196,10); g=Rational(98,10); result = (u**2/(2*g)==Rational(196,10))"},
  {"claim":"Q10 resultant speed = sqrt(34)","code":"result = (sqrt(5**2+3**2)==sqrt(34))"}
]