Introduction to vectors and scalars

Vector = something that has a magnitude and direction.

Scalar = something that has only got magnitude.

Say if we move a block 5 meters. What we are describing is a scalar quantity. If we say that this brick has moved 5m to the left that it becomes a vector quantity (or DISPLACEMENT).

Say if we took 2 seconds to move the brick, then we can say that we moved this brick at 2.5m/s (5 meters / 2 seconds). This is a Scalar quantity.

However, if we give it a direction (we moved this brick 5 meters to the left and it took us 2 seconds) then it becomes a vector quantity.

Introduction to reference frames (whose perspective)

From the ground

• We are stationary
• Plane has velocity to the right (250m/s)
• Car has velocity to the left of 50m/s

From the car

• Car seems stationary.
• The ground looks like its moving at 50m/s
• The plane looks like it’s going faster at 250m/s + 50m/s = 300m/s

From the plane

• The plane seems stationary.
• The ground looks like it’s moving quickly 250m/s to the left.
• The car looks like it’s moving 300m/s

What is displacement?

Displacement = Change in position. Implies that an object has moved, or has been displaced.

Formula: Displacement = Change in x = x (final position) – x (initial position)

Displacement is a vector, it has a direction as well as a magnitude and is represented visually by an arrow. For example, I pace left and right while talking, the 2.0, displacement of myself relative to my desk is represented by an arrow pointing to the right.

My initial position is x (initial) = 1.5m and my final position is x (final) = 3.5m. The displacement is: 3.5m – 1.5m = +2.0m. In this co-ordinate system, motion to the right is positive, while motion to the left is negative.

What do distance and distance travelled mean?

Distance = defined as the magnitude of displacement between two positions. Distance between two positions is not the same as the distance travelled between them.

Distance travelled = the total length of the path travelled between 2 positions. Distance travelled is not a vector.

N.B Distance travelled does not have to equal the magnitude of the displacement. It can be greater.

Calculating average velocity or speed

Let’s use an example here:

If Sam was able to travel 5km north in 1 hour in his car, what was his average velocity?

Velocity = displacement / time = distance / change in time = 5 km / 1 hour = 5/1 = 5 km/h North. (You have to give a direction for a vector quantity).

For m/s: 5km/h = 1000m/1km = 5000m/hour

hour/seconds = 1 hour / 3600 seconds

5km / 1 hour = 5000 m / 3600 seconds = 1.39 m/s

Solving for time

Ben is running at a constant velocity of 3 m/s to the east. How long will it take him to travel 720 meters?

Time = Displacement / Speed

Time = 720m / 3m/s = 240 seconds.

Instantaneous Speed and Velocity

Instantaneous Speed = Speed at a particular moment in time.

Say you start running at 6m/s, then at 2m/s and then finally at 8m/s

Instantaneous Velocity = Velocity at a particular moment in time.

V(average) = change in distance / change in time. This is our formula for instantaneous velocity.

What does velocity mean?

Average velocity is defined to be the change in position divided by he time of trave

V (avg) = ((position(final) – position(initial)) / (time(final) – time(initial))

This definition indicates that velocity is a vector because displacement is a vector. For example:

Suppose a passenger took 5 seconds to move 4 meters to the left (-4m), where displacement is towards the back of the plane, the average velocity can be written as:

V (avg) = -4m / 5 seconds = -0.8 m/s

What does speed mean?

In Physics, speed has no direction. Thus, speed is a scalar. Instantaneous speed is the magnitude of instantaneous velocity. For example:

Suppose our passenger at one instant had an instantaneous velocity 0f -3.0 m/s, at the same time his instantaneous speed was 3.0m/s

Average speed is different from average velocity. Average speed is the distance travelled divided by the elapsed time. So while the size of instantaneous speed and velocity are identical, the size of average speed and velocity can be very different.

Since distance travelled can be greater than displacement, the average speed can be greater than average velocity, which is displacement divided by time. For example:

If you drive to a store and return home in 30 minutes and you have travelled 6km, then your average speed was 12km/hr.

Your displacement is zero, because you have gone and come back, therefore your overall position hasn’t changed. Thus, the average speed is not simply the magnitude of average velocity.

Velocity and Speed Example

A lizard with a poor sense of spatial awareness is walking back and forth in the desert. The lizard walks 12m to the right in a time of 20 seconds, then runs 16m to the left in a time of 8 seconds. The average speed and average velocity of the lizard for the entire trip would be as follows:

average speed = (12.0m + 16.0m) / (20s + 8s) = 28.0m/28.0 s = 1m/s

average velocity = displacement / time = -4.0m/28.0s = -1/7 m/s

Position vs Time graphs

The vertical axis on a position graph represents the position of the object.

The slope on a position graph represents the velocity of the object. The value of the slope at a particular time represents the velocity of the object at that instant.

The slope of a position graph at a given moment in time gives you the instantaneous velocity at that moment in time.

The average slope between two points in time will give you the average velocity between those two points in time.

However, if the slope is constant for a period of time, then the instantaneous velocity will equal the average velocity between only two points on that line segment.

The curvature on a position graph.

• If the position graph is curved, the slope will be changing, which also means the velocity is changing. Changing velocity implies acceleration. Curvature means the object is accelerating.
• If the curvature looks like an upside down bowl, the acceleration is negative.
• If the curvature looks like a right side up bowl, the acceleration is positive