Vectors and Scalars

 

Dynamics is, by definition, a branch of mechanics which focuses on the motion of bodies under the action of forces. To properly understand motion, forces and dynamics in physics, we must open with some definitions to get the ball rolling.

Scalars

Scalars are quantities that are fully described by its magnitude.

Examples:

  • Time

  • Mass

  • Energy

  • Speed

  • Distance

  • Volume

You may say: “This cup has a volume of 1L.” Therefore, volume is fully described by magnitude

Vectors

Vectors are quantities that are described by both magnitude and direction.

Examples:

  • Velocity

  • Displacement

  • Force

  • Acceleration

  • Momentum

You may say: “The car is travelling on the motorway at a velocity of 20m/s, travelling due-East.” So we require a direction to be added to vectors to fully understand the quantity, which in this case is velocity.

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Displacement and Velocity

You may be familiar with the relationship of: Distance = Speed x Time. We are now going to introduce Vector quantities into the mix.

Displacement is a distance with a direction.

Velocity is a speed with a direction.

The relationship therefore can be adapted to become:

Displacement = Average Velocity x Time

Average velocity = Displacement/Time

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Resultant Vectors

When adding vectors, you must take both magnitude and direction into account.

You can do this in two ways:

You can use a scale diagram, by using a ruler to draw out the pathway of the vectors.

  • Say that you need to draw a scale diagram of a bike cycling 80m north and 90m west to calculate the resultant displacement. You equate every centimetre on your ruler to 10m and draw 8cm north, followed by 9cm west, making it a scale diagram.

  • You then can draw a line between your starting and ending points, which would be the resulting displacement.

  • You then can use trigonometry to calculate the direction of the vector. SOHCAHTOA (Sin - opposite/hypotenuse, Cos - Adjacent/hypotenuse, Tan - opposite/adjacent) is always very handy for this.

You can also calculate resultant vectors mathematically, using Pythagoras theorem. This is demonstrated in the given diagram.

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Velocity-Time Graphs

You calculate acceleration by calculating the gradient of the line. Gradient can be calculated just like in maths: m=(y2-y1)/(x2-x1).

You can calculate displacement by calculating the area under the line. It is worth remembering how to calculate the area of a rectangle and the area of a triangle when calculating displacement from a graph.

Area of a triangle = 1/2 base x height

Area of a rectangle = base x height

Key Points!

  • Vectors and Scalars

    You need to be able to understand and define the difference between vectors and scalars, as well as provide examples of the two.

  • Resultant Vectors

    Calculating resultant vectors is a very common exam question which requires multiple steps.

    Easy marks can be picked up here by practicing calculating resultant vectors.

    Use Pythagoras and SOHCAHTOA to calculate vectors mathematically.

  • Displacement, Velocity, Time

    Understanding s = vt and being able to rearrange this equation will help further on in this unit.

    On velocity-time graphs:

    • Gradient is acceleration

    • Area under the graph is displacement

Any Questions?