Physics 9702/11 — May/June 2012

Background Concept

A scalar has magnitude only (e.g. energy, time, mass). A vector has magnitude and direction (e.g. force, displacement, velocity).

Work done is defined by the dot product

$$W = \vec{F} \cdot \vec{s} = Fs\cos\theta$$

where $\theta$ is the angle between $\vec{F}$ and $\vec{s}$. A dot product always produces a scalar.

Understanding the Question

You are given the special-case equation

$$W = Fs$$

and asked how many of the quantities $W$, $F$, and $s$ are vectors and how many are scalars.

The wording “displacement $s$ in the direction of the force” tells you that $\theta = 0^\circ$, so $\cos\theta = 1$.

Approach

1. Classify each quantity: $W$, $F$, $s$.
2. Use the fact that work is energy transferred, which is a scalar.
3. Recall that force and displacement are vectors, even if they happen to be along the same line in this situation.

Step-by-Step Reasoning

• $W$ (work done) is a form of energy transfer, and energy is a scalar.
• $F$ (force) has direction (push/pull in a particular direction), so it is a vector.
• $s$ (displacement) is a change in position with direction, so it is a vector.
• The equation $W = Fs$ is the dot-product result when $\vec{F}$ and $\vec{s}$ are parallel; it does not mean that $F$ and $s$ are scalars.

So there is one scalar ($W$) and two vectors ($F$ and $s$).

Key Takeaways

• Work done is always a scalar.
• Force and displacement are vectors.
• $W = Fs$ is a special case of $W = \vec{F} \cdot \vec{s}$ when the vectors are in the same direction.

Common Mistakes

• Treating $F$ and $s$ as scalars just because the equation is written without vector arrows.
• Thinking that if two vectors are in the same direction they “become scalars” (they do not; they are still vectors).

Things to Be Careful About

• The full definition involves $\cos\theta$; the question explicitly states “in the direction of the force” to justify using $W = Fs$.
• Some symbols can represent either magnitude or vector depending on context; here the physics classification remains: force and displacement are vector quantities.