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].
As the duration of the time interval [math]\displaystyle{ \Delta t }[/math] becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.
Derivative Relationships
Velocity is the time derivative of position:
[math]\displaystyle{ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} }[/math].
Acceleration, in turn, is the time derivative of velocity:
[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} }[/math]
Integral Relationships
Position is the time integral of velocity:
[math]\displaystyle{ \vec{r}(t) = \int \vec{v}(t) \ dt }[/math].
Velocity is, in turn, the time integral of acceleration:
[math]\displaystyle{ \vec{v}(t) = \int \vec{a}(t) \ 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.
In Physics
According to Newton's First Law of Motion, the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. Newton's Second Law: the Momentum Principle describes how velocity changes over time as a result of forces.
A Computational Model
In VPython simulations of physical systems, the Iterative Prediction algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.
Click here for velocity simulation
Click "view this program" in the top left corner to view the source code.
Examples
Simple
A runner runs a lap counterclockwise around a circular track at a constant speed of 2.8m/s.
a) What is the average speed of the runner over the course of the lap?
b) What is the average velocity of the runner over the course of the lap?
c) What is the instantaneous velocity of the runner at the northernmost point on the track?
Solutions:
a) The instantaneous speed of the runner is 2.8m/s for the entire duration of the lap, so the average speed of the runner is also 2.8m/s.
b) The instantaneous velocity vector of the runner has a constant magnitude of 2.8m/s, but its direction is constantly changing depending on the runner's position along the track. To find the average value of the velocity vector over time, let us use the following formula:
[math]\displaystyle{ \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} }[/math].
Because the runner ran a single lap and ended at the point in space where they began, their displacement [math]\displaystyle{ \Delta \vec{r} }[/math] is <0,0>m! The runner's average velocity was therefore <0,0>m/s. This makes sense because the velocity vector spends equal time pointing in every possible direction in the plane of the track, so each instantaneous velocity at any point in the lap cancels with a different instantaneous velocity at a different point in the lap.
c) At the northernmost point on the track, the runner had a speed of 2.8m/s as always and was running in the westward direction because the problem tells us they were running counterclockwise. This is a sufficient answer, as it describes the magnitude and direction of the velocity vector. Alternatively, the answer can be reported in component form: <-2.8,0>m/s using +x to mean east and +y to mean north.
Difficult
a particle moves along the x axis. At time t=0, its position is x=2. Its velocity [math]\displaystyle{ v }[/math] varies over time, obeying the following function:
[math]\displaystyle{ v(t) = \begin{cases} 4-2t & \text{if $t\leq 3$} \\ -2 & \text{if $t\gt 3$} \end{cases} }[/math]m/s
What is its position as a function of time after time t=0?
Solution:
[math]\displaystyle{ x(t) = x(0) + \int_0^t v(t')dt' }[/math] ([math]\displaystyle{ t' }[/math] is a "dummy variable" since [math]\displaystyle{ t }[/math] is already our limit of integration)
for t[math]\displaystyle{ \leq }[/math]3,
[math]\displaystyle{ x(t) = 2 + \int_0^t 4-2t' dt' }[/math]
[math]\displaystyle{ x(t) = 2 + [4t'-t'^2]_0^t }[/math]
[math]\displaystyle{ x(t) = 2 + 4t - t^2 }[/math]
for t>3,
[math]\displaystyle{ x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt' }[/math]
[math]\displaystyle{ x(t) = 5 + \int_3^t -2 dt' }[/math]
[math]\displaystyle{ x(t) = 5 + [-2t']_3^t }[/math]
[math]\displaystyle{ x(t) = 5 -2t + 6 }[/math]
[math]\displaystyle{ x(t) = 11 - 2t }[/math]
final answer:
[math]\displaystyle{ x(t) = \begin{cases} -t^2 + 4t + 2 & \text{if $t\leq 3$} \\ 11-2t & \text{if $t\gt 3$} \end{cases} }[/math]m/s
Difficult
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.
History
See Also
External links
The Physics Classroom: Speed and Velocity
HyperPhysics: Average Velocity
YouTube video explaining average vs instantaneous velocity
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.
