2.1 Scalars and vectors — Practise

Work through these on your own. For each quantity, decide whether a direction is needed before you write anything down.

The questions and the answers below come from a set that is checked by computer every time the book is built.

Classify each quantity as a scalar or a vector: mass, weight, displacement, energy, acceleration, temperature.
A hiker walks $4.2\,\text{km}$ north, then turns and walks $1.7\,\text{km}$ south. Find the resultant displacement.
A car travels $1.2\,\text{km}$ east in $60\,\text{s}$ . Find its average velocity.
Two forces act along the same line. A force of $300\,\text{N}$ acts to the right, and a force of $450\,\text{N}$ acts to the left. Find the resultant force, and state its direction.
Explain the difference between distance and displacement, using a journey that ends where it started.
A plane flies $800\,\text{km}$ east and then $800\,\text{km}$ west. Find its final displacement.
An electric current has a direction, but it is not a vector. Explain why.
A displacement is written as $250\,\text{m}$ on a bearing of $060^\circ$ . State its magnitude and its direction.
A car travels round a roundabout at a steady $8.0\,\text{m}\,\text{s}^{-1}$ . Explain why its speed is constant but its velocity is not.
A trolley moves $2.0\,\text{m}$ to the right, then $5.0\,\text{m}$ to the left. Find its resultant displacement.
Which of these quantities are vectors: force, power, momentum, work, impulse, pressure? Give the magnitude and the direction for one of them.
A ferry points straight across a river at $4.0\,\text{m}\,\text{s}^{-1}$ . The current flows at $3.0\,\text{m}\,\text{s}^{-1}$ at right angles to the crossing. A student says the velocity over the ground is $7.0\,\text{m}\,\text{s}^{-1}$ . Explain why this is wrong, and say what rule gives the correct value.

Answers and concise solutions

1. Scalars and vectors

Mass, energy and temperature are scalars, because each has a magnitude but no direction.
Weight, displacement and acceleration are vectors, because each has a direction as well as a magnitude.

Answer: Mass, energy and temperature are scalars. Weight, displacement and acceleration are vectors.

2. The resultant displacement

Answer: The two displacements are along the same line but in opposite directions, so subtract: $4.2 - 1.7 = 2.5\,\text{km}$ north.

3. The average velocity

Answer: Velocity is displacement divided by time. In metres, $d = 1.2 \times 10^3\,\text{m}$ , so $v = \dfrac{1200}{60} = 20\,\text{m}\,\text{s}^{-1}$ east.

4. The resultant force

Answer: Take right as positive. The resultant is $300 - 450 = -150\,\text{N}$ , so $150\,\text{N}$ to the left.

5. Distance and displacement

Distance is a scalar: it is the total length of the path travelled, with no direction.
Displacement is a vector: it is the straight-line distance and direction from the start to the finish.
For a runner who completes one lap of a 400 m track, the distance is 400 m, but the displacement is zero, because the finish is the same point as the start.

Answer: Distance is the total length of the path travelled. Displacement is the straight-line distance and direction from the start to the finish. A runner who completes one lap of a 400 m track covers a distance of 400 m, but the displacement is zero.

6. The final displacement

Answer: The two displacements are equal and opposite, so the resultant displacement is $0\,\text{km}$ . The distance travelled is $1600\,\text{km}$ , but the plane is back where it started.

7. Why current is not a vector

A vector is a quantity that is added by the triangle rule, so two vectors combine into a single resultant with a direction.
Currents do not combine in that way. Two currents meeting at a junction add as numbers at the junction, and a current is not drawn as an arrow with a length, so current is a scalar.

Answer: A vector must add by the triangle rule. Currents do not combine like that, so current is a scalar even though it has a direction.

8. The magnitude and direction

The magnitude is 250 m.
The direction is a bearing of 060 degrees, measured clockwise from north.

Answer: The magnitude is $250\,\text{m}$ . The direction is the bearing $060^\circ$ , measured clockwise from north.

9. Speed and velocity on a roundabout

The speed is the magnitude of the velocity, and it stays at 8.0 m s^-1, because the car covers the same distance each second.
The velocity is a vector. Its direction changes as the car goes round, so the velocity changes even though its magnitude does not.

Answer: Speed is a scalar and stays at $8.0\,\text{m}\,\text{s}^{-1}$ . Velocity is a vector whose direction changes round the roundabout, so the velocity changes.

10. The resultant displacement

Answer: Take right as positive. The displacement is $2.0 - 5.0 = -3.0\,\text{m}$ , so $3.0\,\text{m}$ to the left.

11. The vector quantities

Force, momentum and impulse are vectors. Power, work and pressure are scalars.
For example, a momentum of 1.5 kg m s^-1 to the north has a magnitude of 1.5 and a direction of due north.

Answer: Force, momentum and impulse are vectors. Power, work and pressure are scalars. A momentum of $1.5\,\text{kg}\,\text{m}\,\text{s}^{-1}$ to the north has magnitude $1.5$ and direction due north.

12. Why the velocities do not simply add

The two velocities are not along the same line, so their magnitudes cannot be added as ordinary numbers. The ferry is carried sideways while it crosses.
The two velocities are at right angles, so the resultant is found with Pythagoras: the square of the resultant is the sum of the squares of the two velocities.

Answer: The velocities are at right angles, not along the same line, so $4.0 + 3.0$ is wrong. The resultant is found from Pythagoras: $v = \sqrt{4.0^2 + 3.0^2} = 5.0\,\text{m}\,\text{s}^{-1}$ .