Velocity and displacement — vectors and scalars

Vector and Scalar Quantities

 

A scalar quantity is fully described by its magnitude (size) and unit, e.g.

A vector quantity is fully described by its magnitude, unit, and direction, e.g.

Distance and Displacement

Distance is a scalar quantity.  It measures the total distance travelled, no matter in which direction.

Displacement is a vector quantity.  It is the length measured from the starting point to the finishing point in a straight line. Its direction must be stated.

Example

A girl walks 3 km due north then turns and walks 4 km due east as shown in the diagram.

Calculate

(a)  The total distance travelled
(b)   The girl’s final displacement relative to her starting position.

Solution

(a)  Total distance             = 3 km + 4 km
                                                    = 7 km

(b)  (to calculate displacement, we need to draw a vector diagram) 

(this solution involves using Phythagoras and trig functions (SOH-CAH=TOA), but you can also solve these types of problems using a scale diagram).

 

Speed and Velocity

Speed is a scalar quantity.  As discussed above, speed is the distance travelled per unit time.

Velocity is a vector quantity (the vector equivalent of speed). Velocity is defined as the displacement per unit time.

Since velocity is a vector, you must state its direction. 

The direction of velocity will be the same as the displacement.

Example

The girl’s walk in the previous example took 2.5 hours.

Calculate

(a)  The average speed
(b)  The average velocity for the walk, both in km/h

Solution

(a)   Distance = 7 km                                               
                Time = 2.5 hours

  

  

  

 

(b)   Displacement = 5 km (053˚)                       
                Time = 2.5 hours

  
 

 

 

(When performing calculations on speed and velocity like the above, it is best to write the equations in words as shown, to avoid confusion with symbols.  This communicates your understanding in the clearest way).