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5.4.6Forces

Vector and Scalar Quantities — AQA GCSE Combined Science Trilogy 8464 - Physics Revision Notes & Topical Questions

5 min read 9 parts Lesson 23 of 44

📚 AQA GCSE Combined Science Trilogy 8464 - Physics Hub / Topical Revision
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Part 1 of 7Scalars and vectors

A scalar quantity has only a size (magnitude). A vector quantity has both a size and a direction.

ScalarsVectors
distance, speed, mass, time, energy, temperaturedisplacement, velocity, acceleration, force, weight, momentum

Saying a car moves at 20 m/s gives a speed; saying it moves at 20 m/s north gives a velocity.

Part 2 of 7Distance and displacement

Distance is the total length of path travelled. Displacement is the straight-line distance from the start to the finish, with a direction. A runner who completes one 400 m lap of a track has travelled a distance of 400 m but has a displacement of zero, because she ends where she began.

Likewise, speed is distance divided by time, but velocity is displacement divided by time. A velocity can be negative if we choose a positive direction and the object moves the other way.

Speed itself can also be an average: average speed = total distance ÷ total time.

1
Because a vector has direction, average velocity = displacement ÷ time.
2
On a round trip, the average velocity is zero even though the average speed is not.
3
A car travelling around a circular track at constant speed has a changing velocity because its direction keeps changing,
  • so it is accelerating even at constant speed.
4
This surprising point is a good example of why the difference matters.

Part 3 of 7Representing vectors

A vector is drawn as an arrow.

The length is proportional to the magnitude, using a scale such as 1 cm = 10 N, and the arrow points in the direction of the vector.
Head to tail
To add vectors we place them head to tail: draw the first, start the second from the tip of the first,
  • and the vector from the start of the first to the tip of the last is the resultant.

Vectors in the same line are easy: forces of 6 N right and 4 N left give 2 N right. For vectors at right angles the resultant is the diagonal of a rectangle.

To subtract vectors, add the opposite vector.

Representing vectors
For example, if a ball moves right at 5 m/s and bounces back left at 5 m/s, its change in velocity is (−5) − (+5) = −10 m/s, a change of 10 m/s to the left even though the speed has not changed.
This idea is used later when finding the change in momentum in collisions.

Vectors can also be added in the opposite order and give the same resultant, which is why it does not matter which arrow you draw first in a scale diagram.

Part 4 of 7Combining perpendicular vectors

Two vectors at right angles form two sides of a right-angled triangle, and the resultant is the hypotenuse.

R2 = A2 + B2

The angle between the resultant and one side comes from tan θ = opposite / adjacent. A scale drawing with a ruler and protractor gives the same result within drawing accuracy, and is accepted in exams when the scale is clearly stated.

For scale drawings choose a scale that makes the diagram fill most of the page, for example 1 cm to 2 N.

Combining perpendicular vectors
Use a sharp pencil, a ruler and a protractor, label each arrow with its size, and write the scale on the diagram.
Measure the resultant length and convert back with the scale.
Check your answer by calculation where the vectors are perpendicular.

Part 5 of 7Resolving a vector (Supplement)

A single vector can be replaced by two perpendicular components.

Resolving a vector (Supplement)
A force F at angle θ above the horizontal has a horizontal component F cos θ and a vertical component F sin θ.
Resolving is useful for objects on slopes or for a rope pulling at an angle.
Each component can then be treated separately, because horizontal and vertical motion do not affect each other.

Part 6 of 7Why it matters

Many quantities in physics depend on direction. The next lesson on resultant forces uses vector addition, and the momentum topics treat momentum as a vector, so a ball bouncing back off a wall changes momentum even if its speed stays the same.

Part 7 of 7Worked examples

  1. A student walks 30 m east then 40 m north. Find the distance walked and the displacement.

  2. A boat heads across a river at 3.0 m/s while the current flows at 4.0 m/s along the river. Find the boat's speed relative to the bank.

  3. Sort the following: mass, velocity, energy, force, speed.

Key terms

TermMeaning
ScalarA quantity with magnitude only.
VectorA quantity with magnitude and direction.
DisplacementStraight-line distance from start to end with direction.
ResultantThe single vector that has the same effect as two or more vectors.
ComponentOne of two perpendicular parts of a vector.

Avoid these mistakes

1
Using distance and displacement as if they mean the same thing.
2
Adding vectors at right angles by simple addition (3 + 4 = 7) instead of using Pythagoras.
3
Saying speed and velocity are identicalvelocity needs a direction.
4
Forgetting the arrowhead and direction when stating a vector answer.
5
Thinking energy has a direction because it can be transferred in a direction.

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