Exercise 10.1 · Q3: Scalar or vector: time period, distance, force, velocity, work done

Basics you need

Part (i): time period

1. Decide and classify

   A time period is the time taken for one complete cycle of a repeating motion, for example a pendulum that takes 2 seconds to swing there and back. "2 seconds" says everything; time does not point in any direction in space.

   So time period is a scalar.

   A time period is a length of time, and time is in the book's list of scalars. Example 2 (i) classes "5 seconds" as Time, a scalar, for the same reason.
   Time is a scalar quantity (Section 10.1; Example 2 (i)). Basics: some answers need a "which way"

Part (ii): distance

1. Compare distance with displacement

   Walk 3 m east from A to B, then 4 m north from B to C. The distance travelled is the length of the path, $3 + 4 = 7$ m. The displacement is the single arrow $\overrightarrow{AC}$⁠, of length 5 m, pointing from A to C.

   The distance only adds up lengths, and lengths are numbers; no direction survives in "7 m". The displacement needs a direction to say where you ended up. The same trip gives two different quantities, one of each kind.
   The length of $\overrightarrow{AC}$ is found by Pythagoras: $\sqrt{3^2 + 4^2} = 5$⁠. Basics: use Pythagoras

2. Classify

   Distance has magnitude only: distance is a scalar.

   Distance is one of the scalar quantities listed in Section 10.1, while displacement is listed among the vectors. The question asks about distance, so the answer is scalar.
   Distance is a scalar quantity; displacement is a vector quantity (Section 10.1). Basics: some answers need a "which way"

Part (iii): force

1. Decide and classify

   A force is a push or a pull, and it always acts in some direction. A push of 10 newton to the left and a push of 10 newton to the right have the same size but opposite effects: together they cancel.

   So force is a vector.

   The book's opening example is a football pass: the force involves muscular strength (magnitude) and the direction in which the other player stands. Both are needed to describe the force.
   Force is a vector quantity (Section 10.1; Example 2 (iii): "10 Newton" is Force, a vector). Lesson: a vector is a length with a direction

Part (iv): velocity

1. Compare velocity with speed

   Speed says how fast, for example 30 km/hr. Velocity says how fast and which way, for example 20 m/s towards north. Two cars at 60 km/hr on the same road, one going north and one going south, have the same speed but different velocities.

   Velocity is the rate of change of displacement, and displacement has a direction, so velocity inherits it. Speed is the rate of change of distance, a scalar, so speed is a scalar.
   Example 2 (iv) and (vi): "30 km/hr" is Speed, a scalar; "20 m/s towards north" is Velocity, a vector. Lesson: Examples 1–3

2. Classify

   Velocity has magnitude (the speed) and direction: velocity is a vector.

   Velocity is in the list of vector quantities in Section 10.1. Its magnitude is the speed, which is why speed alone is a scalar.
   A quantity that has magnitude as well as direction is a vector (Definition 1, Section 10.2). Basics: some answers need a "which way"

Part (v): work done

1. See what work done is made of

   When a constant force $\vec F$ moves an object through a displacement $\vec d$⁠, the work done is

   $$W = |\vec F|\,|\vec d|\cos\theta,$$

   where $\theta$ is the angle between $\vec F$ and $\vec d$⁠. For example, a 10 N pull at 60° to a 5 m path does work $10 \times 5 \times \cos 60^\circ = 25$ joules.

   Force and displacement are both vectors, but the work is the product of two magnitudes and a cosine, all real numbers, so the result is a real number. Only the part of the force along the path, $|\vec F|\cos\theta$⁠, does work.
   This has the form of the scalar (or dot) product $\vec F\cdot\vec d$ defined in Section 10.6.1 (Definition 2), whose value is a real number. Lesson: the dot product

2. Classify

   "25 joules" says everything about the work; it has no direction. So work done is a scalar.

   Work is in the book's list of scalar quantities (Section 10.1). Energy put into an object does not point anywhere: lifting a box or pushing it along a floor can both use 25 joules.
   Work is a scalar quantity (Section 10.1). Basics: some answers need a "which way"

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