Scalars & Vectors

Module 1 deals with the core foundations of physics, probably the most useful aspects of physics to real life – mechanics.

Scalar & Vectors
Definition of a scalar: A quantity with size (magnitude) only.
Examples: speed, distance, time, mass, volume, work done, pressure, energy, power and density.

Definition of a vector: A quantity with size (magnitude) and direction.
Examples: displacement, velocity, force, weight, gravitational field strength and acceleration.

Displacement is the vector alternative for distance and is defined as distance moved in a particular direction. Velocity in turn is the vector alternative for speed and is defined as speed in a stated direction.


Take the following example:
Tom and Amy are taking a walk in the park. They follow the winding pathway and walk a total of 500m. Their distance is 500m, however their displacement is only be 300m west to east.

Vectors Extended

When two forces act on an object it is important which direction these forces are acting. These forces can often be resolved to find a resultant vector which describes the subsequent movement of the body.

Example
The following two forces can be resolved using the triangle method or the parallelogram method.

Note that vector diagrams always use arrows for forces and the resultant force is represented using a double arrow.

Vector Triangle
When there is no resultant force a vector triangle can be used to show the body is at equilibrium.

Resolving Vectors
In the same way that two vectors can be represented as one resultant vector, it is possible to resolve one vector into two components both at right angles. This is used in particular in the next section concerning projectiles.

 

 

 

Some fundamental equations

 

Graphs

In mechanics, graphs can be used to great effect. There are four main types of graphs you will need to be able to construct and interpret Velocity-Time graphs, Speed-Time graphs, Displacement-Time graphs and Distance-Time graphs.

· For velocity-time graphs the gradient = acceleration and the area under the curve = displacement
· For speed-time graphs the area under the curve = distance

· For displacement-time graphs the gradient = velocity
· For distance-time graphs the gradient = speed

 

Distance-time graph

Example
A cyclist was riding in the park at a constant speed of 3 ms-1 for 12 seconds until he came to a stop for 15 seconds to catch his breath. He then rode back to the entrance in 10 seconds. Calculate his speed during the third stage of his journey by plotting a distance-time graph.

The first step is to draw a distance-time graph.


Now using the information provided for the first stage of the journey it is possible to calculate d.

Gradient = speed
3 =
d = 36 m

It took him 10 seconds to ride back to the entrance. We now using this information need to calculate his speed.

Speed = ms-1