Module 1 — Pre-Main Problem Set

How to use this set

These come before the formal mechanics course, and deliberately so. You are not expected to know a large collection of physics formulas. Each problem gives you enough physical information to begin. Your task is to recognise the mathematical structure inside the physical situation.

For each problem, ask three questions before writing anything:

  1. What physical quantity is being described?

  2. What mathematical object represents it?

  3. What mathematical operation does the question require?

Do not begin by searching for a formula. There are no answers on this page — if you cannot tell whether yours is right, that is worth knowing, and it is a different problem from being wrong.

1. Can the formula be physical?

A student writes the displacement of a particle as x = ut + ⅓a²t³, with u an initial velocity, a an acceleration and t a time. Without computing anything, decide whether this can represent a displacement. If not, say exactly which term fails and why.

2. Finding a hidden exponent

The time T of an oscillation depends only on a length L and an acceleration g, as T = C L^ag^b with C dimensionless. Find a and b from consistency alone, and state what dimensions C must have.

3. Three competing quantities

A quantity is proposed as Q = A²B/C, where A has units m s⁻¹, B has m s⁻² and C has m² s⁻². Find the units of Q, and say which of velocity, acceleration, length or a pure number it could be. Argue from the units rather than choosing.

4. Distance versus displacement

A particle goes from (0, 0) to (3, 4) metres along some curved path. Find the magnitude of its displacement. Can the distance travelled be determined from what you are told? Say why or why not.

5. Two journeys, one endpoint

Particle A travels 10 m east and then 10 m west. Particle B does not move at all. Compare their distances travelled and their displacements. What does the comparison show about path quantities and endpoint quantities?

6. Average velocity

A particle goes from x = 2 m to x = 14 m in 4 s, possibly moving back and forth on the way. Find its average velocity. Can its average speed be found from this information?

7. Cartesian to polar

A point is at (−4, 4√3). Find its polar coordinates with 0 ≤ θ < 2π, then express r̂ and θ̂ at that point in terms of î and ĵ.

8. Same point, different language

A point has r = 10 and θ = 5π/6. Find its Cartesian coordinates. A student obtains (5√3, −5). Without redoing the whole calculation, identify what they have done wrong.

9. A point on a circle

A point moves with x = R cos θ and y = R sin θ, with R constant. Show directly that x² + y² = R². What does that equation describe, and why is this a convenient way to write circular motion?

10. Two simultaneous motions

A boat moves at 4 m/s east relative to the water; the water moves at 3 m/s north relative to the ground. Find the boat’s velocity relative to the ground, its magnitude and its direction.

11. Resolving an oblique vector

A force of 20 N makes 30° with the positive x-axis. Find its components. A second force has components (−10, 10√3) N. Find the resultant of the two.

12. Cancelling a vector

With A = 3î + 4ĵ and B = −5î + 2ĵ, find C such that A + B + C = 0. What does a zero resultant mean geometrically about the three vectors?

13. Minimum correction

A particle has velocity 6î + 8ĵ. A correction u is added so that the final velocity is purely horizontal. Find u and its magnitude. How much of the original velocity did you need to examine?

14. A force along a displacement

A constant force F = 6î + 8ĵ N acts through a displacement d = 5î m. Compute F · d and say what it means physically. What happened to the 8 N?

15. When does a force do no work?

A force aî + 4ĵ acts through a displacement 3î − 2ĵ. Find the value of a for which the work done is zero, and state the geometric relationship that then holds.

16. Projection

Project A = 5î + 12ĵ onto the direction of B = 3î + 4ĵ. Find both the scalar and the vector projection. Check that your scalar answer is not larger than |A|, and say why it could not be.

17. A hidden angle

Two vectors of magnitudes 10 and 6 have a dot product of 30. Find the angle between them. Now suppose the dot product were −30 instead — what would change about their relative directions?

18. Turning effect of a force

A force 5ĵ N acts at r = 2î m from a chosen origin. Compute r × F, giving both magnitude and direction.

19. Same force, different origin

A force 6ĵ N acts at Q = (3, 0, 0) m. Find its torque about the origin and about P = (−2, 0, 0) m. The force did not change. Explain why the torque did.

20. When is the torque zero?

Find the condition on the angle between r and F for which r × F vanishes. Then describe two physically different situations that satisfy it.

21. Vector area

For A = 2î + 3ĵ and B = 4î − ĵ, compute |A × B|. What geometric quantity is this?

22. Position function

A particle has x(t) = t² − 4t + 3. Find x at t = 0, 2 and 4, sketch the graph, and identify the time of minimum position. Do it without differentiating.

23. A velocity graph

A particle has v = 2t for 0 ≤ t ≤ 3, then v = 12 − 2t for 3 < t ≤ 6. Find the displacement from t = 0 to 6 by reading areas, without integrating.

24. Sign matters

A particle’s velocity is negative from t = 0 to 2 and positive from t = 2 to 5. The signed areas under the velocity–time graph are −6 m and +10 m. Find the net displacement, and the total distance if it can be found.

25. Velocity from position

A particle has x(t) = t³ − 6t² + 11t − 4. Find v(t) and a(t), and every instant at which the particle is momentarily at rest.

26. Maximum height

A particle has y(t) = 20t − 5t². Find when y is greatest and that greatest value, and explain why setting dy/dt = 0 is the right move here.

27. A changing rate

A particle has x(t) = t⁴ − 4t³ + 6t². Find every time at which the velocity is zero, and decide for each whether the position has a maximum, a minimum, or neither. Count carefully — there may be fewer than you expect.

28. Acceleration without position

A particle has v(t) = 3t² − 12t + 9. Find the acceleration and the time at which it vanishes. What is happening to the velocity at that instant? It is not at rest.

29. Reconstructing velocity

A particle has a(t) = 4t − 6, with v = 5 m/s at t = 0. Find v(t) and the velocity at t = 3 s. Say what the constant of integration represented physically.

30. Reconstructing position

A particle has v(t) = 3t² − 6t + 4, with x = 2 m at t = 0. Find x(t) and the displacement between t = 1 s and t = 3 s.

31. Distance versus displacement

A particle has v(t) = t² − 4t + 3 over 0 ≤ t ≤ 4. Find the displacement and the total distance. You will need the times at which the direction changes before you can do the second.

32. Area under a changing force

A force varies with position as F(x) = 2x + 3 newtons. Find the work done from x = 0 to x = 5 m, and describe what the integral means geometrically. Check it by computing the area of a trapezium.

33. Small-angle approximation

Estimate sin 5° using sin θ ≈ θ. Compare with the true value and express the percentage error. Remember what units θ must be in.

34. Pendulum approximation

The pendulum obeys d²θ/dt² + (g/L) sin θ = 0. Show what the equation becomes for small θ, and explain what changes about the kind of equation you now have.

35. First-order approximation

Use (1 + x)ⁿ ≈ 1 + nx to estimate (1.02)⁵, and compare with the exact value. For what size of x would you expect this to become unreliable?

36. Approximation with physical meaning

For |x| « 1, find a first-order approximation to √(1 + x), and apply it at x = 0.04. Explain why it should work well there.

37. The complete vector motion  — Foundations 2, 4, 7, 8

A particle has r(t) = (3t² − 2t)î + (4t − 1)ĵ. Find its position at t = 0 and t = 2, the displacement between them and its magnitude, the average velocity over that interval, and the instantaneous velocity and acceleration. Say which of these needed calculus and which did not.

38. Force, displacement and work  — Foundations 2, 4, 5

A particle moves from r₁ = 2î − ĵ to r₂ = 7î + 3ĵ metres under a constant force F = 4î + 2ĵ N. Find the displacement and its magnitude, the work done, and the angle between force and displacement.

39. Motion, reversal and total distance  — Foundations 7, 8, 9

A particle has a(t) = 4t − 10 m/s², with v(0) = 6 m/s and x(0) = 3 m. For 0 ≤ t ≤ 4 s, find v(t), the times at which the particle is at rest, x(t), the displacement, and the total distance. Do not attempt the distance before you know the direction of motion in each interval.

40. The approximation behind circular motion  — Foundations 3, 7, 8, 10

A particle moves on a circle of radius R with θ = ωt, so that x = R cos ωt and y = R sin ωt. Find the velocity and acceleration components, show that the acceleration points toward the centre, and find its magnitude. Explain why this happens even though the speed never changes. Finally, show that for a small angular displacement the arc length and the chord length agree to first order.

When you have finished

Go back to the three questions at the top and apply them to the problems you found hardest. In most cases the difficulty will turn out to have been at step two — knowing what physical quantity was involved but not which mathematical object represents it.

Several of these can be answered with no calculation at all if you see the structure first. Problems 5, 13, 16, 20 and 22 each have a part of that kind. Once this set is comfortable, move on to Problem Set 1.

Covers: Foundations 1–10  ·  Type: Problems, no answers  ·  Difficulty: Pre-Main — read the Foundations first, mechanics not required