Velocity
This page defines and describes velocity.
Main Idea
Velocity, denoted by the symbol [math]\displaystyle{ \vec{v} }[/math], is a vector quantity defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as momentum and kinetic energy, are functions of velocity. The most commonly used metric unit for velocity is the meter per second (m/s).
A Mathematical Model
Instantaneous velocity [math]\displaystyle{ \vec{v} }[/math] is defined as:
[math]\displaystyle{ \vec{v} = \frac{d\vec{r}}{dt} }[/math]
where [math]\displaystyle{ \vec{r} }[/math] is a position vector and [math]\displaystyle{ t }[/math] is time.
Average Velocity
Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted [math]\displaystyle{ \vec{v}_{avg} }[/math]. The average acceleration over an interval of time [math]\displaystyle{ \Delta t }[/math] is given by
[math]\displaystyle{ \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} }[/math].
Derivative Relationships
Acceleration is the time derivative of velocity, which is in turn the time derivative of position, so acceleration is the second time derivative of position.
Recall that
[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} }[/math]
and
[math]\displaystyle{ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} }[/math].
Therefore:
[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2} }[/math].
Integral Relationships
The integral of acceleration yields velocity, and the integral of velocity yields position, so the double integral of acceleration yields position.
[math]\displaystyle{ \vec{v}(t) = \int \vec{a}(t) \ dt }[/math]
and
[math]\displaystyle{ \vec{r}(t) = \int \vec{v}(t) \ dt }[/math].
Therefore
[math]\displaystyle{ \vec{r}(t) = \iint \vec{a}(t) \ dt \ dt }[/math].
Kinematic Equations
The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the Kinematics page.
Newton's Second Law of Motion
In physics, the acceleration of particles is caused by forces. Newton's Second Law: the Momentum Principle states that
[math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math].
From this relation, it can be derived that
[math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math]
, or that
[math]\displaystyle{ \vec{a} = \frac{\vec{F}_{net}}{m} }[/math].
This makes sense because acceleration should be directly proportional to force, which causes it, and inversely proportional to mass, which resists it (see Inertia). This equation is often used to find the acceleration of a particle. If that acceleration is constant, kinematic equations are then often used to determine how its position and velocity change over time.
A Computational Model
This model defines an object and models its displacement arbitrarily with its mass in relation to time,
m=9 g=9.81 t=0 deltat=1 positionInitial=vector(0,0,0) while t<6: positionFinal=vector(0,m*g*t,0) displacement=positionFinal-positionInitial velocity=displacement/deltat t=t+deltat print (velocity ,"is velocity")
after t=6 is reached, the updated velocity is given.
This specific vPython example illustrates how a "while loop" may be used to update velocity, as velocity can be ever-changing as forces are applied.
Example
A car takes 3 hours to make a 230-mile trip from Point A to Point B.
| Hour 1 | Hour 2 | Hour 3 | |
|---|---|---|---|
| Velocity | 80 mph north | 90 mph north | 60 mph north |
There are two kinds of velocity in which one must consider: instantaneous velocity and average velocity. [1] does a good job explaining the difference between the two types of velocity
Instantaneous Velocity
Instantaneous velocity is the speed and direction of an object at a particular instant. Mathematically, it is the derivative of the position function at a specific point in time.
Given the example: Each hour, and each time point in every hour has a different instantaneous velocity.
Average Velocity
Average velocity is the net displacement of an object, divided by the total travel time. It is the average of all instantaneous velocities. It is important to note that as [math]\displaystyle{ {\Delta\mathit{t}} }[/math] gets very small, the average velocity approaches the instantaneous velocity.
Given the example: The average velocity would be (230 miles/3 hours) = 76.67 mph north.
Acceleration
Acceleration is the rate of change of velocity, divided by the change in time, modeled with with the following equation:
- [math]\displaystyle{ \boldsymbol{a} = \frac{\Delta\boldsymbol{v}}{\Delta\mathit{t}} }[/math] ,
where [math]\displaystyle{ {\Delta\boldsymbol{v}} }[/math] is the change of velocity of the object and [math]\displaystyle{ {\Delta\mathit{t}} }[/math] is the change of time.
The SI units for acceleration are meters per second squared (m/s/s). It is also a vector quantity.
Given the example: The acceleration from the 1st hour to the 2nd hour is 10 mph. This indicates a positive acceleration. The acceleration from the 2nd hour to the 3rd hour is -30 mph. This indicates a negative acceleration.
Colloquially acceleration is referred to as "speeding up" whilst "slowing down" is decelerating. Bear in mind that the direction does not have to change for deceleration to take place, it simply has to slow down.
Another Example
Based on what you know about velocity in relation to acceleration. During the time interval of 3-5 seconds is the object accelerating or decelerating? How about from 12-14 seconds? How do you know both of these answers?
Given the Example: From 3-5 seconds, knowing that acceleration is the derivative of velocity, it can be seen that the object is accelerating, as the graph has a positive slopes.From 12-14 seconds, the graph has an increasingly negative slope, signifying deceleration towards zero.
Momentum
Another application of velocity is within the realm of momentum and the Momentum Principle. momentum is defined as the mass of an object multiple by its vector velocity quantity. Like velocity momentum is a vector quantity. This quantity can be used in conjunction with change in time to see the amount of force applied on an object, and by extension its final location and velocity. This can be modeled iteratively through computer programs or be done in one calculation.
Some Examples
An Introductory Example
If a ball travels from location <2,4,6>m to <3,5,8>m in two seconds, what is its velocity?
Solution: :[math]\displaystyle{ \boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{r}}{\Delta\mathit{t}} }[/math] Delta r: <3,5,8>m-<2,4,6>m and delta t is equal to 2s, so velocity is equal to the vector <1,1,2>m/2s, which is equal to <0.5,0.5,1> m/s
A More Difficult Example
A car is moving with a velocity of <26,87,12> m/s. If the initial location of the car is at <0,0,0>m and the final location of the car is at <39,130.5, 18> m, how many seconds did the car travel?
Assuming a constant velocity, delta r is equal to the final location of the car, since the car began at the origin. By dividing the delta r by the given velocity, a total time of two seconds of travel can be found.
A Final Example with Application of Momentum
If a van has a mass of 1200 kilograms, and it is traveling with a velocity of magnitude 38 m/s, what is its momentum?
Its momentum is 45,600 kg*m/s. This can be obtained by multiplying the mass by the magnitude of the velocity.
Connectedness
Velocity is a very simple yet interesting concept in the way that it can be applied to many different parts of physics from something as simple as displacement. Velocity's sheer versatility as a concept and the number of things that can be derived from it, which include acceleration, momentum, and by extension, force and mass. Also because it can be related to force, it can be used, in conjunction with other types of forces to determine many things about systems.
Velocity is also of critical importance in calculating kinetic energy (0.5*m*v^2). Through understanding both the relationship between kinetic energy and velocity, as well as the resulting relationship between kinetic energy and total energy, concepts such as potential spring, gravitational, and electric energies may also be related back to velocity. As Physics 1 deals primarily with motion and predicting future motion, velocity is an absolutely critical tool, as it can often be used to analyze changes of external forces and predict future movement.
Velocity relates to my career aspirations in a rather interesting way. Because I plan on trying to become a trauma doctor, its easy to see the difference between high and low velocity impacts of objects of the same mass. If a low mass object is accelerating at a high enough velocity, the ramifications of its impact with the body could be vastly different than an object with a low velocity.
An industrial application of velocity could be seen in cars and the limits of their engines. The limit to which a car engine can perform can be tested in various ways, one of them being velocity. This could be one reason why you don't see normal cars with speed past around 130, the engine simply can't take it. The knowledge of the limit of a car engine can be tested using velocity to help ensure a safe driving experience for many.
See Also
References
1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.
3. Velocity Expression. Digital image. Physics-Formulas. N.p., n.d. Web. 29 Nov. 2015.
4. Velocity vs Time Graph. Digital image. https://upload.wikimedia.org/wikipedia/commons/a/ae/Velocity-time_graph_example.png. N.p., n.d. Web. 29 Nov. 2015.
