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Unlock This CourseEvery quantity you measure in physics has a size. The size is a number with a unit, such as 20 kg or 45 s. For some quantities the size tells you everything. For others you also need to know which way the quantity points.
Key terms:
Mass is a scalar. A bag of rice has a mass of 2 kg, and it makes no sense to ask "2 kg in which direction?". Force is different. If you only say "a force of 50 N acts on the door", nobody knows whether the door will open or shut. You need the direction as well.
Textbooks often use the word magnitude for size. It means the same thing.
Learn the examples in this table. A common exam question gives a list of quantities and asks you to pick out the scalars or the vectors.
| Scalars (size only) | Vectors (size and direction) |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Weight |
| Time | Force |
| Energy | Acceleration |
Momentum is also a vector. You will meet it in the lesson Momentum.
Notice the two pairs at the top of the table. Each pair has a scalar and a vector that are measured in the same unit.
A postman walks 300 m east along a street, then 300 m west back to his van.
The distance he has walked is 600 m. Distance is how far you have travelled in total, whatever the route.
His displacement is 0 m, because he has ended up where he started. If he had stopped at the end of the street, his displacement would be 300 m east.
Remember, velocity is speed in a given direction.
Two buses each travel at 15 m/s. One heads north and the other heads south. They have the same speed but different velocities: 15 m/s north and 15 m/s south.
To give a vector fully, always write a size and a direction.
Weight is in the vector column because weight is a force. It always acts downwards, towards the centre of the Earth. Mass has no direction, so it is a scalar.
A force has a size, measured in newtons, and a direction. So force is a vector quantity. This is why forces are drawn as arrows:
An arrow for a 40 N force should be twice as long as an arrow for a 20 N force. In fact, any vector can be drawn as an arrow in this way.
Because forces have direction, you cannot just add their sizes together. Two forces of 10 N can add up to 20 N, or they can cancel out completely. It depends on which way each one points.
Most objects have more than one force acting on them. The resultant force is the single force that has the same effect as all the forces acting on an object added together. Other names for it are the net force and the unbalanced force.
In this course you only need to find the resultant of forces that act along a line, such as left and right, or up and down. There are two rules:
A resultant force is a vector too, so your answer needs a size, a unit and a direction.
Two people push a broken-down car along a level road. One pushes forwards with a force of 250 N and the other pushes forwards with a force of 300 N. Ignore friction. Calculate the resultant force.
The forces act in the same direction, so add them.
Resultant force = 250 + 300 = 550 N
Resultant force = 550 N forwards
In a tug of war, team A pulls the rope to the left with a force of 1800 N. Team B pulls to the right with a force of 1650 N. Calculate the resultant force on the rope.
The forces act in opposite directions, so subtract.
Resultant force = 1800 − 1650 = 150 N
Team A's force is larger, so the resultant force is 150 N to the left.
A shopper pushes a loaded trolley forwards with a force of 60 N. Friction of 25 N and air resistance of 5 N both act backwards. Calculate the resultant force on the trolley.
Step 1: add up the forces in each direction.
Total forwards = 60 N
Total backwards = 25 + 5 = 30 N
Step 2: subtract the smaller total from the larger.
Resultant force = 60 − 30 = 30 N
Resultant force = 30 N forwards
Another way to keep track is to choose one direction as positive and the opposite direction as negative. For the trolley, call forwards positive: (+60) + (−25) + (−5) = +30 N. The plus sign tells you the resultant is forwards. A negative answer would mean it acts backwards.
If the forces on an object cancel out, the resultant force is zero. The forces are balanced.
The velocity of the object does not change. If it is at rest, it stays at rest. If it is moving, it keeps moving at a constant velocity: the same speed in the same direction.
If the forces do not cancel out, there is a resultant force. The forces are unbalanced.
The velocity of the object changes. It speeds up, slows down or changes direction, depending on which way the resultant force acts.
A helicopter hovering in one place is a good example of balanced forces. Its weight of 30 000 N acts downwards and the lift from its blades is 30 000 N upwards. The resultant force is 30 000 − 30 000 = 0 N, so it stays at rest in the air.
The same is true for an aeroplane cruising at a steady speed in a straight line at a constant height. The thrust from its engines is equal to the drag, and the lift from its wings is equal to its weight, so the resultant force is zero. The aeroplane is moving fast, but its velocity is not changing.
How large the change in velocity is for a given resultant force is covered in the lesson Force, Mass and Acceleration.
A snowmobile travels in a straight line across a level, snowy field. Its engine gives a driving force of 1500 N forwards. Friction of 400 N and air resistance of 300 N both act backwards.
(a) State what is meant by a vector quantity. [1 mark]
(b) Which one of these quantities is a scalar: acceleration, energy, force, velocity? [1 mark]
(c) Calculate the resultant force on the snowmobile. State its direction. [3 marks]
(d) Later, the driving force is 900 N, and the friction and air resistance add up to 900 N. Describe the motion of the snowmobile. Give a reason for your answer. [2 marks]
(a) A quantity that has both size (magnitude) and direction. (1)
(b) Energy. (1)
(c) Total backwards force = 400 + 300 = 700 N (1)
Resultant force = 1500 − 700 = 800 N (1)
The resultant force acts forwards. (1)
(d) It moves at a constant velocity (steady speed in a straight line). (1) The forces are balanced, so the resultant force is zero. (1)
In part (c) one mark is for the direction alone, so never stop at the number. In part (d), do not write that the snowmobile stops. It is already moving, and a zero resultant force means its velocity stays the same. Only an unbalanced force could slow it down.