- Characteristics of the mean velocity vector v
- The signs of average speed
- Average speed: a scalar quantity
- Exercise resolved
- References
The average velocity for a moving particle is defined as the ratio between the variation of the position that it experiences and the time interval used in the change. The simplest situation is one in which the particle moves along a straight line represented by the x-axis.
Suppose that the moving object occupies the positions x 1 and x 2 at times t 1 and t 2 respectively. The definition of average velocity v m is represented mathematically like this:
The units of v m in the International System are meters / second (m / s). Other commonly used units that appear in texts and mobile devices are: km / h, cm / s, miles / h, ft / s, and more, as long as they are of the form length / time.
The Greek letter "Δ" is read "delta" and is used to briefly indicate the difference between two quantities.
Characteristics of the mean velocity vector v
Average speed is an important characteristic of movement. Source: Pixabay
Average velocity is a vector, since it is related to the change in position, which in turn is known as the displacement vector.
This quality is represented in bold or by an arrow above the letter that designates the magnitude. However, in one dimension, the only possible direction is that of the x-axis and therefore the vector notation can be dispensed with.
Since vectors have magnitude, direction, and sense, an initial look at the equation indicates that the average velocity will have the same direction and sense as the displacement.
Let's imagine the particle in the example moving along a straight line. To describe its movement, it is necessary to indicate a reference point, which will be the "origin" and will be denoted as O.
The particle can move toward or away from O, either to the left or to the right. It can also take a short or long time to reach a certain position.
The magnitudes that have been mentioned: position, displacement, time interval and average speed, describe the behavior of the particle while it moves. It is the kinematic quantities.
To distinguish the positions or locations to the left of O, the sign (-) is used and those to the right of O carry the sign (+).
The average speed has a geometric interpretation that can be seen in the following figure. It is the slope of the line that passes through points P and Q. When cutting the curve position vs. time at two points, it is a secant line.
Geometric interpretation of the average velocity, as the slope of the line joining points P and Q. Source: す じ に く シ チ ュ ー.
The signs of average speed
For the following analysis, it must be taken into account that t 2 > t 1. That is, the next instant is always greater than the current one. In this way t 2 - t 1 is always positive, which usually makes sense on a daily basis.
Then the sign of the mean velocity will be determined by that of x 2 - x 1. Note that it is important to be clear about where the point O -the origin- is, since this is the point with respect to which the particle is said to go "to the right" or "to the left".
Either "forward" or "backward", as the reader prefers.
If the mean velocity is positive, it means that on average the value of "x" increases over time, although this does not mean that it may have decreased at some point in the time period considered - Δt -.
However in global terms, at the end of time Δt, she ended up with a larger position than she had at the beginning. The details of the movement are ignored in this analysis.
What if the average speed is negative? Then it means that the particle ends with a smaller coordinate than the one with which it started. Roughly he moved back. Let's look at some numerical examples:
Example 1: Given the indicated starting and ending positions, indicate the sign of the average speed. Where did the particle move globally?
a) x 1 = 3 m; x 2 = 8 m
Answer: x 2 - x 1 = 8 m - 3 m = 5 m. Positive mean velocity, the particle moved forward.
b) x 1 = 2 m; x 2 = -3 m
Answer: x 2 - x 1 = -3 m - 2 m = -5 m. Negative mean velocity, the particle moved backwards.
c) x 1 = - 5 m; x 2 = -12 m
Answer: x 2 - x 1 = -12 m - (-5 m) = -7 m. Negative mean velocity, the particle moved backwards.
d) x 1 = - 4 m; x 2 = 10 m
Answer: x 2 - x 1 = 10 m - (-4m) = 14 m. Positive mean velocity, the particle moved forward.
Can the average speed be 0? Yes. As long as the starting point and the arrival point are the same. Does this mean that the particle was necessarily at rest the entire time?
No, it just means that the trip was roundtrip. Perhaps it traveled quickly or perhaps very slowly. For now it is not known.
Average speed: a scalar quantity
This leads us to define a new term: average speed. In Physics it is important to distinguish between vector quantities and non-vector quantities: scalars.
For the particle that made the round trip, the average velocity is 0, but it may or may not have been very fast. To find out, the average speed is defined as:
The units for average speed are the same as those for average speed. The fundamental difference between the two quantities is that the average velocity includes interesting information about the direction and direction of the particle.
Instead, the average speed provides only numerical information. With it, it is known how fast or slow the particle moved, but not whether it moved forward or backward. So it is a scalar quantity. How to distinguish them when denoting them? One way is by leaving the bold for the vectors, or by placing an arrow on them.
And it is important to note that the average speed does not have to be equal to the average speed. For the round trip the average speed is zero, but the average speed is not. Both have the same numerical value when you always travel in the same direction.
Exercise resolved
You drive back home from school leisurely at 95 km / h for 130 km. It starts to rain and slows down to 65 km / h. She finally gets home after driving for 3 hours and 20 minutes.
a) How far is your home from the school?
b) What was the mean speed?
Answers:
a) Some preliminary calculations are necessary:
The trip is divided into two parts, the total distance is:
d = d1 + d 2, with d1 = 130 km
t2 = 3.33 - 1.37 hours = 1.96 hours
Calculation of d 2:
d 2 = 65 km / hx 1.96 h = 125.4 km.
The school is d1 + d 2 = 255.4 km from the house.
b) Now the mean speed can be found:
References
- Giancoli, D. Physics. Principles with Applications. Sixth Edition. Prentice Hall. 21-22.
- Resnick, R. (1999). Physical. Volume 1. Third edition in Spanish. Mexico. Compañía Editorial Continental SA de CV 20-21.
- Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7 ma. Edition. Mexico. Cengage Learning Editors. 21-23.