Velocity — Average Velocity, Instantaneous Velocity, and Speed
Calculating Average Velocity or Speed
If Shanatu travels 5km North in 1 hour, what was his average velocity?
$\vec { v } =\sfrac { \vec {s} }{ t }$
In this case, which is actually fairly interesting, the average deonotes in this case both average direction and velocity. The ‘s’ is used for displacement because d is used in Calculus to symbolize the derivative function.
Solving For Time
Ex. If Ben is running at a constant velocity of 3 m/s to the east, how long will it take him to travel 720 meters?
To solve this question we use the same expression as before relating velocity to displacement and time.
\[
\vec { v } =\frac { \vec { s } }{ t } \quad \quad t=\frac { \vec {s} }{ \vec {v} }
\]
Then we rearrange the expression to isolate time (t0 and can then divide our displacement, the 720 meters that we desire this object to travel by the velocity in meters/second that Ben is travelling at, so that we may arrive at our desired value, which turns out to be 4 minutes. At the velocity that Ben is “running” at (for this can hardly be called running, considering fast football players and other athletes can run 40 meters in about 4.5 seconds, moving only 3 meters in a second will be incredibly slow), it will take Ben 240 seconds or 4 minutes to travel 720 meters.
Displacement from Time and Velocity
Ex. If a car is travelling at 45 km/h (kilometers per hour) for 26 minutes, how far will that car be displaced in 26 minutes.
The first step in solving a question like this is…
Instantaneous Speed and Velocity
This is a useful if you have a body that is moving at a variable speed, such as the planet Mercury which moves far faster when it is closer to the Sun – instantaneous speed and velocity is determining the speed or velocity of an object at a specific point in time even if it has a variable speed.
Instantaneous speed therefore is an objects speed at a precise moment in time, its direction at that moment gives you the objects instantaneous velocity, which is different from average velocity, which is the average velocity over the entire trip.
To find the instantaneous velocity you will want to figure out the objects displacement over smaller and smaller Δt’s until you are approaching 0, which means that your changes in time in which you are measuring displacement are getting infinitismally small.
Smaller displacement/shorter time intervals
The problem is that if you wanted a perfect instantaneous velocity, you would ultimately be dividing 0 by 0, which was a problem for a long time, which made it seem like defining motion at any momebt in time seem impossible so that some ancient Greeks questioned whether motion had any meaning at all, of it was not just an illusion.
However, it makes sense that you can only approach zero, the moment, because if time was truly zero then you would have no time elapse, which means that you would have had no rate of change in which movement could have occurred at all. It wasn’t until Newton invented calculus, a whole new branch of mathematics just to deal with this problem. Its answer might look something like this:
[Equation in Notebook 1, figure 1.] To solv this without calculus you could look at the slope of a displacement vs time graph, or of acceleration is constant, you can use kinematics formulas.
What is Velocity? Velocity is an objects change in position divided by the time of travel, in some unity of distance and of time.
\[
\vec { v } =\frac {\Delta x}{\Delta t}=\frac {{ x }_{ f } – {x}_{0}} {{t}_{f} – {t}_{0}}
\]
If the starting time is taken to be zero, then:
\[
{ v }_{ avg } =\frac { \Delta x }{ \Delta t }
\]
This definition indicates that velocity is a vector because it has both magnitude and direction. The International System of Units (SI) has meters per second (m/s) the standard unit for velocity, ut km/h or cm/s are also common.