Answer Key – Word Sheet (Units, Dimensions & Kinematics)

Prepared: August 11, 2025

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MCQs (Q1–36): Answers with Brief Explanations

1. (d) Torque and Energy — both have dimensions of work, [ML²T⁻²].

2. No incorrect option — all four conversions shown are actually correct; likely a misprint in the question.

3. (d) 0.2 mm — 20 VSD = 16 MSD ⇒ 1 VSD = 0.8 mm; L.C. = 1 − 0.8 = 0.2 mm.

4. (d) Work — kg·m·s⁻² is a newton (force), not work (joule).

5. (d) 3 — 0.0690 has 6, 9, and the trailing 0 significant → 3 s.f.

6. (b) 4.00 — more significant figures ⇒ greater precision.

7. (c) a + b — absolute errors add for A − B.

8. (b) Principle of homogeneity — LHS & RHS of any valid equation share dimensions.

9. (c) Distance — a light-year is a length.

10. (b) Kelvin — SI base unit of temperature.

11. (a) 10 m/s — 36 × 1000 / 3600 = 10.

12. (c) Tension and surface tension — tension [MLT⁻²] vs surface tension [MT⁻²] (force/length).

13. (a) force/area — definition of pressure.

14. (d) Light year — not a time unit.

15. (c) 1 — 0.007 has only the 7 significant.

16. (b) Universal gravitational constant — a true dimensional constant.

17. (b) 14% — add % errors with powers: 3·1 + 2·2 + 1·3 + 1·4 = 14.

18. (a) 7% — ΔR/R = ΔV/V + ΔI/I = 5/100 + 0.2/10 = 0.07.

19. (a) 10⁶ — 10⁻³ s / 10⁻⁹ s = 10⁶.

20. (c) Magnetic field — tesla is derived, not fundamental.

21. (b) [ML²T⁻³] — power = work/time.

22. (d) [F v⁻¹ T] — from F = M v / T ⇒ M = F v⁻¹ T.

23. (b) [ML⁻¹T⁻²] — stress = force/area.

24. (b) [ML⁵T⁻²] — a has units P·V².

25. (d) Velocity of light in vacuum — universal constant with dimensions.

26. (a) [F L⁻⁴ T²] — ρ = M/L³ and M = F L⁻¹ T².

27. (d) Coefficient of viscosity — [ML⁻¹T⁻¹].

28. (a) frequency — h/I ∼ (ML²T⁻¹)/(ML²) = T⁻¹.

29. (c) surface energy and surface tension — both = energy/area = force/length.

30. (b) relative density — dimensionless and unitless.

31. (d) — statement is incorrect because percentage error is just relative error ×100; not fundamentally “different.”

32. (a) — random error reduces by taking many readings & averaging.

33. (b) 129.6 unit — 1 UF = 1 quintal·km·h⁻² = 0.007716 N ⇒ 1 N ≈ 129.6 UF.

34. (a) — horizontal velocity component stays constant (no air drag).

35. (c) 1:1 — R ∝ sin 2θ; sin 60° = sin 120°.

36. (a) increasing with time — if s ∝ t³, then a ∝ t.

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1-mark Questions: Model Answers

1. 4 s.f. (0.004560 → 4560 with 4 significant digits).

2. 5.38×10³ (3 s.f.).

3. A×B = 2.3×10⁻² (limited to 2 s.f. by B).

4. 2.50×10⁻⁴ m (3 s.f.).

5. 4 s.f. (7.300, trailing zeros after decimal count).

6. 0.0098760 (5 s.f.) has more than 98760 (4 s.f.).

7. 3 decimal places (rule for addition: least decimal places).

8. 0.205% ((0.5/243.6)×100).

9. 1.5 (units) (absolute error |98.5−100.0|).

10. 2% (least-count 0.1/5.0×100).

11. 0.3 m (errors add for x+y).

12. 2% ((0.1/5.0)×100).

13. SI (International System of Units).

14. A base unit defined by convention (e.g., m, kg, s).

15. kelvin (K).

16. watt (W).

17. Measuring large distances (e.g., stellar) by apparent shift.

18. Nearest power of 10 of a value.

19. Impulse: [MLT⁻¹] (same as momentum).

20. Yes (e.g., ball at top of its flight: v=0, a=−g).

21. Positive acceleration (curve bends upward on x–t).

22. Net displacement = 0 (areas cancel).

23. Average velocity = 0 (overall displacement zero).

24. Uniform circular motion (speed const, direction changes).

25. ×4 (since s ∝ v²).

26. v_y = 0, a_y = −g at the highest point.

27. Speed decreases (accel opposite to velocity).

28. No — one speeds up, the other slows down.

29. 14.7 m in the next second (s₂−s₁ = ½ g (4−1)).

30. Straight line (x–t with constant slope).

31. Constant negative acceleration (downward-sloping straight v–t).

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2-mark Questions: Concise Solutions

1. Motion follows velocity’s direction, not acceleration; e.g., stone thrown up has downward a but moves upward initially.

2. Frame-dependent: at rest in one frame, moving in another (train example).

3. Same height (if same u): H = u²/2g, independent of mass.

4. At top of projectile: v=0 but a=−g (also car turning with v≠0, a≠0).

5. Relative velocity is zero when they have same velocity vectors (equal speed & direction).

6. Avoid systematic (constant) error by calibration, method correction.

7. Numerical value changes; physical quantity doesn’t (unit conversion).

8. Near star shows greater parallax for same baseline.

9. Examples: semiconductors, lasers, MRI, satellites.

10. e.g., C.V. Raman, S.N. Bose, H.J. Bhabha, S. Chandrasekhar.

11. 10,000 quintals in a gigagram.

12. 1 amu = 1.66×10⁻²⁷ kg.

13. 3.9×10⁸ m (distance = ct/2 with t = 2.6 s).

14. ≈2.16% (sum % errors of length and breadth).

15. 1% ((0.05/5)×100).

16. 3.9732×10¹⁶ m (4.2 ly).

17. ≈2.06×10⁵ AU.

18. ≈5.48×10⁻⁴ amu (9.1×10⁻³¹ / 1.66×10⁻²⁷).

19. Order =10⁹ (since 5.67×10⁸ > √10×10⁸).

20. Volume ≈ 1.08×10²¹ m³; order = 10²¹.

21. Avg speed = total distance/time; instant speed = limit at an instant (speedometer).

22. x = x₀ + vt + ½at² (write all three eqs with x₀ ≠ 0).

23. Straight line v–t: uniform acceleration.

24. Upward positive ⇒ g is negative.

25. At top: v=0, a=−g.

26. Stopping distance = u²/(2|a|) (from v²−u²=2as).

27. Area under v–t gives displacement.

28. a = dv/dt; slope of v–t at a point.

29. Example: top of vertical throw (v=0, a=−g).

30. Slope of x–t graph = velocity.

31. Zero (constant speed in 1-D ⇒ a=0).

32. x=5t²+3t ⇒ v=10t+3; at t=3, v=33 m/s.

33. u=20, v=0, t=4 ⇒ a=−5 m/s², s=40 m.

34. u=72 km/h = 20 m/s ⇒ s=50 m.

35. 700 m (200 m while accelerating + 500 m at constant speed).

36. Final v=25 m/s, displacement 100 m.

37. a=−1.25 m/s², s=30 m.

38. t=4 s.

39. 170 m/s (Galilean addition 150+20).

40. 62.5 m.

41. u=19.6 m/s, H=19.6 m (total time = 4 s).

42. 1100 m total.

43. v=22 m/s, s=128 m.

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3-mark Questions: Succinct Working / Definitions

1. Mean length = 2.504 m; mean absolute error ≈ 0.0088 m.

2. Relative error = |9.8−10.0|/10.0 = 0.02 ⇒ 2%.

3. Mean of totals = 20.46 s; per oscillation = 2.046 s; % error vs 2.05 s ≈ 0.195%.

4. For [M¹L²T⁻³]: max % error = 1(ΔM%) + 2(ΔL%) + 3(ΔT%) = 5 + 8 + 15 = 28%.

5. Principle of homogeneity: each term of a valid equation has same dimensions. For E=½mv²: [E]=[M][LT⁻¹]²=[ML²T⁻²] (OK).

6. Systematic: constant bias (e.g., zero-error) → correct via calibration. Random: scatter about mean → reduce by averaging. Least-count: resolution-limited (e.g., 0.1 s stopwatch).

7. Accuracy = closeness to true value; precision = repeatability. Precise but not accurate: readings tightly clustered yet all shifted.

8. SI advantages: universality, coherence (derived from base), decimal multiples (prefixes), defined standards, interoperability.

9. Oleic-acid film method: spread known volume on water, measure patch area after spreading to monolayer; d = V/A (with dilution & assumptions).

10. Fundamental vs derived units (e.g., m, kg, s vs N, J); fundamentals are the independent building blocks.

11. Good standard unit: invariant, accessible, reproducible, well-defined—ensures comparability.

12. CGS/FPS/MKS/SI comparison; SI preferred for coherence & global adoption.

13. Order of magnitude: nearest power of 10; examples: AU/ly/parsec vs atomic sizes.

14. Define AU, ly, parsec; relation 1 pc ≈ 3.26 ly.

15. Mass: scalar, additive, invariant (non-relativistic); note at high v: m = γ m₀ (concept).

16. Acceleration: rate of change of velocity; positive vs negative with examples.

17. Graphical derivation of v=u+at from area under a–t or slope of v–t.

18. Instant speed vs velocity: scalar magnitude vs vector; both at an instant.

19. Area under v–t = displacement (integral).

20. Sign of a indicates speeding up/slowing down relative to v.

21. Upwards throw: v↓ to 0, a=−g always; downward: v↑, a=−g.

22. Displacement vs path length (vector vs scalar) with everyday example.

23. Free fall: motion under gravity alone; v=u+gt, s=ut+½gt², v²−u²=2gs.

24. u=20, t=4 ⇒ a=−5 m/s², s=40 m.

25. From rest with a=5: in 8 s, s=160 m.

26. u=15 m/s: H = u²/2g ≈ 11.5 m; t↑ = u/g ≈ 1.53 s.

27. 36 km/h = 10 m/s, a=0.5: after 40 s, v=30 m/s.

28. u=25, t=5: a=−5, s=62.5 m.

29. Drop from 100 m: t≈4.52 s, v≈44.3 m/s.

30. Through 10 cm plank: u=200, v=100, s=0.10 m ⇒ a=(v²−u²)/(2s)=−1.5×10⁵ m/s².

31. Using s₅=50 m and s₁₀=150 m ⇒ u=5 m/s, a=2 m/s².

32. From u=10, a=2, after s=100: v=√(u²+2as)=√220 ≈ 14.83 m/s.

33. u=20, t=4 ⇒ a=−5, s=40 m.

34. u:5→15 in 10 s: a=1 m/s², s=100 m.

35. H=45 m ⇒ u=√(2gH) ≈ 29.7 m/s.

36. 72 km/h=20 m/s, stop in 10 s: a=−2 m/s², s=100 m.

here’s a clean answer key (with quick reasons) for the MCQs (Q1–36) from your worksheet.

1. (d) Torque and Energy — both have dimensions of work, $[ML^2T^{-2}]$.
2. No incorrect option — all four conversions shown are actually correct; likely a misprint in the question.
3. (d) 0.2 mm — $20\ \text{VSD} = 16\ \text{MSD}\Rightarrow 1\ \text{VSD}=0.8\ \text{mm}$; L.C. $=1-0.8=0.2\ \text{mm}$.
4. (d) Work — $\text{kg·m·s}^{-2}$ is a newton (force), not work (joule).
5. (d) 3 — 0.0690 has 6,9, and the trailing 0 significant → 3 s.f.
6. (b) 4.00 — more significant figures ⇒ greater precision.
7. (c) $a+b$ — absolute errors add for $A-B$.
8. (b) Principle of homogeneity — LHS & RHS of any valid equation share dimensions.
9. (c) Distance — a light-year is a length.
10. (b) Kelvin — SI base unit of temperature.
11. (a) 10 m/s — $36\times1000/3600=10$.
12. (c) Tension and surface tension — tension $[MLT^{-2}]$ vs surface tension $[MT^{-2}]$ (force/length).
13. (a) force/area — definition of pressure.
14. (d) Light year — not a time unit.
15. (c) 1 — 0.007 has only the 7 significant.
16. (b) Universal gravitational constant — a true dimensional constant.
17. (b) 14% — add % errors with powers: $3·1+2·2+1·3+1·4=14$.
18. (a) 7% — $\Delta R/R=\Delta V/V+\Delta I/I=5/100+0.2/10=0.07$.
19. (a) $10^{6}$ — $10^{-3}\text{ s}/10^{-9}\text{ s}=10^{6}$.
20. (c) Magnetic field — tesla is derived, not fundamental.
21. (b) $[ML^{2}T^{-3}]$ — power = work/time.
22. (d) $[Fv^{-1}T]$ — from $F= M v/T \Rightarrow M=Fv^{-1}T$.
23. (b) $[ML^{-1}T^{-2}]$ — stress = force/area.
24. (b) $[ML^{5}T^{-2}]$ — $a$ has units $P·V^{2}$.
25. (d) Velocity of light in vacuum — universal constant with dimensions.
26. (a) $[FL^{-4}T^{2}]$ — $\rho=M/L^{3}$ and $M=FL^{-1}T^{2}$.
27. (d) Coefficient of viscosity — $[ML^{-1}T^{-1}]$.
28. (a) frequency — $h/I\sim (ML^{2}T^{-1})/(ML^{2})=T^{-1}$.
29. (c) surface energy and surface tension — both $=$ energy/area = force/length.
30. (b) relative density — dimensionless and unitless.
31. (d) — statement is incorrect because percentage error is just relative error ×100; not fundamentally “different.”
32. (a) — random error reduces by taking many readings & averaging.
33. (b) 129.6 unit — $1\ \text{UF}=1\ \text{quintal·km·h}^{-2}=0.007716\ \text{N}\Rightarrow 1\ \text{N}\approx129.6\ \text{UF}$.
34. (a) — horizontal velocity component stays constant (no air drag).
35. (c) 1:1 — $R\propto\sin2\theta$; $\sin60^\circ=\sin120^\circ$.
36. (a) increasing with time — if $s\propto t^{3}$, then $a\propto t$.

awesome — here’s a crisp, ready-to-use answer key (with brief reasons/steps) for the 1-mark, 2-mark, and 3-mark sections of your worksheet.

1-mark answers (with quick reasons)

1. 4 s.f. (0.004560 → 4560 with 4 significant digits).

2. $5.38\times10^3$ (3 s.f.).

3. $A\times B=2.3\times10^{-2}$ (limited to 2 s.f. by $B$).

4. $2.50\times10^{-4}\ \text{m}$ (3 s.f.).

5. 4 s.f. (7.300, trailing zeros after decimal count).

6. 0.0098760 (5 s.f.) has more than 98760 (4 s.f.).

7. 3 decimal places (rule for addition: least decimal places).

8. $0.205\%$ $(0.5/243.6)\times100$.

9. 1.5 (units) (absolute error $|98.5-100.0|$).

10. 2% (least-count $0.1/5.0\times100$).

11. 0.3 m (errors add for $x+y$).

12. 2% $(0.1/5.0)\times100$.

13. SI (International System of Units).

14. A base unit defined by convention (e.g., m, kg, s).

15. kelvin (K).

16. watt (W).

17. Measuring large distances (e.g., stellar distances) by apparent shift.

18. Nearest power of 10 of a value.

19. Impulse: $[MLT^{-1}]$ (same as momentum).

20. Yes (e.g., ball at top of its flight: $v=0, a=-g$).

21. Positive acceleration (curve bends upward on $x$-$t$).

22. Net displacement = 0 (areas cancel).

23. Average velocity = 0 (overall displacement zero).

24. Uniform circular motion (speed const, direction changes).

25. ×4 (since $s\propto v^2$).

26. $v_y=0,\ a_y=-g$ at the highest point.

27. Speed decreases (accel opposite to velocity).

28. No—one speeds up, the other slows down (depends on $a$ vs $v$).

29. 14.7 m in the next second $(s_{2}-s_{1}=\tfrac12 g(4-1))$.

30. Straight line ($x$–$t$ with constant slope).

31. Constant negative acceleration (downward-sloping straight $v$–$t$).

2-mark answers (concise steps & results)

1. Motion follows velocity’s direction, not acceleration; e.g., stone thrown up has downward $a$ but moves upward initially.

2. Frame-dependent: at rest in one frame, moving in another (train example).

3. Same height (if same $u$): $H=u^2/2g$, independent of mass.

4. At top of projectile: $v=0$ but $a=-g$ (also car turning with $v\ne0$, $a\ne0$).

5. Zero when they have same velocity vectors (equal speed & direction).

6. Avoid systematic (constant) error by calibration, method correction.

7. Numerical value changes; physical quantity doesn’t (unit conversion).

8. Near star shows greater parallax for same baseline.

9. Examples: semiconductors, lasers, MRI, satellites.

10. e.g., C.V. Raman, S.N. Bose, H.J. Bhabha, S. Chandrasekhar.

11. 10,000 quintals in a gigagram.

12. $1\ \text{amu}=1.66\times10^{-27}\ \text{kg}$.

13. $3.9\times10^8\ \text{m}$ (distance $=ct/2$ with $t=2.6\ \text{s}$).

14. $\approx2.16\%$ (sum % errors of length and breadth).

15. 1% $(0.05/5)\times100$.

16. $3.9732\times10^{16}\ \text{m}$ (4.2 ly).

17. $\approx2.06\times10^5\ \text{AU}$.

18. $9.1\times10^{-31}/1.66\times10^{-27}\approx5.48\times10^{-4}\ \text{amu}$.

19. Order $=10^9$ (since $5.67\times10^8>\sqrt{10}\times10^8$).

20. Volume $\approx1.08\times10^{21}\ \text{m}^3$; order $=10^{21}$.

21. Avg speed = total distance/time; instant speed = limit at an instant (speedometer).

22. $x=x_0+vt+\tfrac12at^2$ (write all three eqs with $x_0\neq0$).

23. Straight line $v$-$t$: uniform acceleration.

24. Upward positive ⇒ $g$ is negative.

25. At top: $v=0,\ a=-g$.

26. Stopping distance $=u^2/(2|a|)$ (derive from $v^2-u^2=2as$).

27. Area under $v$-$t$ gives displacement.

28. $a=\dfrac{dv}{dt}$; slope of $v$-$t$ at a point.

29. Example: top of vertical throw ($v=0$, $a=-g$).

30. Slope of $x$-$t$ graph = velocity.

31. Zero (constant speed in 1-D ⇒ $a=0$).

32. $x=5t^2+3t\Rightarrow v=10t+3$; at $t=3$, $v=33\ \text{m/s}$.

33. $u=20$, $v=0$, $t=4\Rightarrow a=-5\ \text{m/s}^2$, $s=40\ \text{m}$.

34. $u=72\ \text{km/h}=20\ \text{m/s}\Rightarrow s=50\ \text{m}$.

35. 700 m (200 m while accelerating + 500 m at constant speed).

36. Final $v=25\ \text{m/s}$, displacement $100\ \text{m}$.

37. $a=-1.25\ \text{m/s}^2$, $s=30\ \text{m}$.

38. $t=4\ \text{s}$.

39. $170\ \text{m/s}$ (Galilean addition $150+20$).

40. $62.5\ \text{m}$.

41. $u=19.6\ \text{m/s}$, $H=19.6\ \text{m}$ (total time $=4\ \text{s}$).

42. $1100\ \text{m}$ total.

43. $v=22\ \text{m/s},\ s=128\ \text{m}$.

3-mark answers (succinct working / definitions)

1. Mean length $=2.504\ \text{m}$; mean absolute error $\approx 0.0088\ \text{m}$.

2. Relative error $=|9.8-10.0|/10.0=0.02$; 2%.

3. Mean of totals $=20.46\ \text{s}$; per oscillation $=2.046\ \text{s}$; % error vs $2.05\ \text{s}$ $\approx0.195\%$.

4. For $[M^1L^2T^{-3}]$: max % error $=1(\%\Delta M)+2(\%\Delta L)+3(\%\Delta T)=5+8+15=28\%$.

5. Principle of homogeneity: each term of a valid equation has same dimensions. For $E=\tfrac12 mv^2$: $[E]=[M][LT^{-1}]^2=[ML^2T^{-2}]$ (OK).

6. Systematic: constant bias (e.g., zero-error) → correct via calibration. Random: scatter about mean → reduce by averaging. Least-count: resolution-limited (e.g., 0.1 s stopwatch).

7. Accuracy = closeness to true value; precision = repeatability. Precise but not accurate: readings tightly clustered yet all shifted.

8. SI advantages: universality, coherence (derived from base), decimal multiples (prefixes), defined standards, interoperability.

9. Oleic-acid film method: spread known volume on water, measure patch area after spreading to monolayer; $d=V/A$ (with dilution & assumptions).

10. Fundamental vs derived units (e.g., m, kg, s vs N, J); fundamentals are the independent building blocks.

11. Good standard unit: invariant, accessible, reproducible, well-defined—ensures comparability.

12. CGS/FPS/MKS/SI comparison; SI preferred for coherence & global adoption.

13. Order of magnitude: nearest power of 10; examples: AU/ly/parsec vs atomic sizes.

14. Define AU, ly, parsec; relation $1\ \text{pc}\approx3.26\ \text{ly}$.

15. Mass: scalar, additive, invariant (non-relativistic); at high $v$: $m=\gamma m_0$ (concept note).

16. Acceleration: rate of change of velocity; positive vs negative with examples.

17. Graphical derivation of $v=u+at$ from area under $a$-$t$ or slope of $v$-$t$.

18. Instant speed vs velocity: scalar magnitude vs vector; both at an instant.

19. Area under $v$-$t$ = displacement (integral).

20. Sign of $a$ indicates speeding up/slowing down relative to $v$.

21. Upwards throw: $v\downarrow$ to 0, $a=-g$ always; downward: $v\uparrow$, $a=-g$.

22. Displacement vs path length (vector vs scalar) with everyday example.

23. Free fall: motion under gravity alone; $v=u+gt,\ s=ut+\tfrac12gt^2,\ v^2-u^2=2gs$.

24. $u=20,\ t=4\Rightarrow a=-5\ \text{m/s}^2,\ s=40\ \text{m}$.

25. From rest with $a=5$: in 8 s, $s=160\ \text{m}$.

26. $u=15\ \text{m/s}$: $H=\dfrac{u^2}{2g}\approx11.5\ \text{m}$; $t_\uparrow=\dfrac{u}{g}\approx1.53\ \text{s}$.

27. $36\ \text{km/h}=10\ \text{m/s}$, $a=0.5$: after $40\ \text{s}$, $v=30\ \text{m/s}$.

28. $u=25,\ t=5$: $a=-5$, $s=62.5\ \text{m}$.

29. Drop from $100\ \text{m}$: $t\approx4.52\ \text{s}$, $v\approx44.3\ \text{m/s}$.

30. Through 10 cm plank: $u=200,\ v=100,\ s=0.10\ \text{m}\Rightarrow a=\dfrac{v^2-u^2}{2s}=-1.5\times10^5\ \text{m/s}^2$.

31. Uses $s_5=50\ \text{m}$ and $s_{10}=150\ \text{m}\Rightarrow u=5\ \text{m/s},\ a=2\ \text{m/s}^2$.

32. From $u=10,\ a=2$, after $s=100$: $v=\sqrt{u^2+2as}=\sqrt{20+200}= \sqrt{220}\approx14.83\ \text{m/s}$.

33. $u=20$, $t=4\Rightarrow a=-5,\ s=40\ \text{m}$.

34. $u:5\to15$ in 10 s: $a=1\ \text{m/s}^2$, $s=100\ \text{m}$.

35. $H=45\ \text{m}\Rightarrow u=\sqrt{2gH}\approx29.7\ \text{m/s}$.

36. $72\ \text{km/h}=20\ \text{m/s}$, stop in 10 s: $a=-2\ \text{m/s}^2,\ s=100\ \text{m}$.

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