FHSST Physics/Vectors/Examples

Some Examples of Vectors

Displacement

Imagine you walked from your house to the shops along a winding path through the veld. Your route is shown in blue in Figure 3.1. Your sister also walked from the house to the shops, but she decided to walk along the pavements. Her path is shown in red and consisted of two straight stretches, one after the other.

Figure 3.1: Illustration of Displacement

Although you took very different routes, both you and your sister walked from the house to the shops. The overall effect was the same! Clearly the shortest path from your house to the shops is along the straight line between these two points. The length of this line and the direction from the start point (the house) to the end point (the shops) forms a very special vector known as displacement. Displacement is assigned the symbol ${\overrightarrow {s}}$

Definition: Displacement is defined as the magnitude and direction of the straight line joining one's starting point to one's final point.

OR

Definition: Displacement is a vector with direction pointing from some initial (starting) point to some final (end) point and whose magnitude is the straight-line distance from the starting point to the end point.

(NOTE TO SELF: choose one of the above)

In this example both you and your sister had the same displacement. This is shown as the black arrow in Figure 3.1. Remember displacement is not concerned with the actual path taken. It is only concerned with your start and end points. It tells you the length of the straight-line path between your start and end points and the direction from start to finish. The distance travelled is the length of the path followed and is a scalar (just a number). Note that the magnitude of the displacement need not be the same as the distance travelled. In this case the magnitude of your displacement would be considerably less than the actual length of the path you followed through the veld!

Velocity

Definition: Velocity is the rate of change of displacement with respect to time.

The terms rate of change and with respect to are ones we will use often and it is important that you understand what they mean. Velocity describes how much displacement changes for a certain change in time.

We usually denote a change in something with the symbol $\Delta$ (the Greek letter Delta). You have probably seen this before in maths — the gradient of a straight line is ${\frac {\Delta y}{\Delta x}}$. The gradient is just how much y changes for a certain change in x. In other words it is just the rate of change of y with respect to x. This means that velocity must be

${\begin{matrix}{\overrightarrow {v}}={\frac {\Delta {\overrightarrow {s}}}{\Delta t}}={\frac {{\overrightarrow {s}}_{final}-{\overrightarrow {s}}_{initial}}{t_{final}-t_{initial}}}\end{matrix}}$

(NOTE TO SELF: This is actually average velocity. For instantaneous $\Delta$'s change to differentials. Explain that if $\Delta$ is large then we have average velocity else for infinitesimal time interval instantaneous!)

What then is speed? Speed is how quickly something is moving. How is it different from velocity? Speed is not a vector. It does not tell you which direction something is moving, only how fast. Speed is the magnitude of the velocity vector (NOTE TO SELF: instantaneous speed is the magnitude of the instantaneous velocity.... not true of averages!).

Consider the following example to test your understanding of the differences between velocity and speed.

Worked Example 3: Speed and Velocity

Question: A man runs around a circular track of radius 100m. It takes him 120s to complete a revolution of the track. If he runs at constant speed, calculate:

1. his speed,
2. his instantaneous velocity at point A,
3. his instantaneous velocity at point B,
4. his average velocity between points A and B,
5. his average velocity during a revolution.

Answer:

1. To determine the man's speed, we need to know the distance he travels and how long it takes. We know it takes $120s$ to complete one revolution of the track. What distance is one revolution of the track? We know the track is a circle and we know its radius, so we can determine the perimeter or distance around the circle. We start with the equation for the circumference of a circle:

${\begin{matrix}C&=&2\pi r\\&=&2\pi (100m)\\&=&628.3\;m.\end{matrix}}$

2. Now that we have distance and time, we can determine speed. We know that speed is distance covered per unit time. If we divide the distance covered by the time it took, we will know how much distance was covered for every unit of time.

${\begin{matrix}v&=&{\frac {Distance\ travelled}{time\ taken}}\\&=&{\frac {628.3m}{120s}}\\&=&5.23\ m.s^{-1}\end{matrix}}$

3. Consider point A in the diagram:

We know which way the man is running around the track, and we know his speed. His velocity at point A will be his speed (the magnitude of the velocity) plus his direction of motion (the direction of his velocity). He is moving at the instant that he arrives at A, as indicated in the diagram below.

His velocity vector will be $5.23\ m.s^{-1}$ West.

4. Consider point B in the diagram:

We know which way the man is running around the track, and we know his speed. His velocity at point B will be his speed (the magnitude of the velocity) plus his direction of motion (the direction of his velocity). He is moving at the instant that he arrives at B, as indicated in the diagram below.

His velocity vector will be $5.23\ m.s^{-1}$ South.

4. So, now, what is the man's average velocity between Point A and Point B?

As he runs around the circle, he changes direction constantly. (Imagine a series of vector arrows pointing out from the circle, one for each step he takes.) If you add up all these directions and find the average it turns out to be ...Right. South west. And, notice that if you just looked for the average between his velocity at Point A and at Point B, that comes out south west, too. So his average velocity between Point A and Point B is $5.23\ m.s^{-1}$ south west.

5. Now we need to calculate his average velocity over a complete revolution. The definition of average velocity is given earlier and requires that you know the total displacement and the total time. The total displacement for a revolution is given by the vector from the initial point to the final point. If the man runs in a circle, then he ends where he started. This means the vector from his initial point to his final point has zero length. A calculation of his average velocity follows:

${\begin{matrix}{\overrightarrow {v}}&=&{\frac {\Delta {\overrightarrow {s}}}{\Delta t}}\\&=&{\frac {0m}{120s}}\\&=&0\ m.s^{-1}\end{matrix}}$

Remember: Displacement can be zero even when distance is not!

Acceleration

Definition: Acceleration is the rate of change of velocity with respect to time.

Acceleration is also a vector. Remember that velocity was the rate of change of displacement with respect to time so we expect the velocity and acceleration equations to look very similar. In fact:

${\begin{matrix}{\overrightarrow {a}}={\frac {\Delta {\overrightarrow {v}}}{\Delta t}}={\frac {{\overrightarrow {v}}_{final}-{\overrightarrow {v}}_{initial}}{t_{final}-t_{initial}}}\end{matrix}}$

(3.2)

(NOTE TO SELF: average and instantaneous distinction again! expand further — what does it mean?)

Acceleration will become very important later when we consider forces.

Force

Imagine that you and your friend are pushing a cardboard box kept on a smooth floor. Both of you are equally strong. Can you tell me in which direction the box will move ? Probably not. Because I have not told you in which direction each of you are pushing the box. If both of you push it towards north, the box would move northwards. If you push it towards north and you friend pushes it towards east, it would move north-eastwards. If you two push it in opposite directions, it wouldn't move at all !

Thus in dealing with force applied on any object, it is equally important to take into account the direction of the force, as the magnitude. This is the case with all vectors.