Velocity

CLAIMED BY ANJIYA KARIM (FALL 2022)

Main Idea

Velocity is a vector quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol [math]\displaystyle{ \vec{v} }[/math] or v, as opposed to [math]\displaystyle{ v }[/math], which denotes speed. Velocity is 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 magnetic force, are functions of velocity. Velocity is given in unit distance per unit time. The SI 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 velocity 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]

where [math]\displaystyle{ \Delta \vec{r} }[/math] is the displacement (change in position) over that time interval ([math]\displaystyle{ \Delta \vec{r} = \vec{r}_f - \vec{r}_i }[/math]).

Average velocity is often confused with average speed. A section on the Speed vs Velocity page goes into depth about the difference.

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.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity [math]\displaystyle{ \vec{v}_i }[/math] and final velocity [math]\displaystyle{ \vec{v}_f }[/math]:

[math]\displaystyle{ \vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2} }[/math]

Unlike the first equation, this equation is only true if acceleration is constant.

Instantaneous Velocity

Instantaneous velocity is defined as the velocity of an object in motion at a specific point in time. Determining instantaneous velocity is similar to determining the average velocity. However, in instantaneous velocity, we shorten the period of time over which we measure the change in distance to a point where the time has approached zero. While instantaneous velocity is defined as the velocity of a body at a specific point in time, the average velocity is the displacement over the time taken for the body to displace itself. Visually, this can be seen as the following:

Graph Displaying Relationship Between Instantaneous and Average Velocity

The instantaneous velocity can be found by using calculus and taking the derivate of a distance vs time function at a specific point in time. If the position function is f(t), then the velocity at a time t would be v(t) = f'(t).

Angular Velocity

Angular velocity, also known as rotational velocity, is how fast a body rotates around a specific center of rotation. It is denoted as omega ω. There are two ways an object can rotate:

       1) Rotating around its own center of mass. For example, spinning a ball on your finger.
       2) Rotating around a point in space. For example, the rotation of Earth around the Sun. 

The higher the angular velocity, the faster the object would rotate. In order to determine the its direction, we can use the right hand rule. By curling the fingers of the right hand in the direction in which the object is turning, the thumb would end up pointing in the direction of the axis of rotation.

Angular Velocity Formulas

  There are two formulas for calculating angular velocity:
       1) ω=(α2−α1)/t, where α represents angles of a circle in radians and t represents time in seconds.
       2) ω=r x v/|r|^2, where v represents velocity, and r represents the radius of the circle.

Calculating Angular Velocity:

   We can calculate the average angular velocity of the moon in its orbit around the Earth by the following method:
       -The moon takes about 27.3 days (2,358,720 seconds)to orbit around the Earth
       -ω=2π/t = 2.7*10^-6/s.

Velocity and Acceleration Relationship

Similar to how the velocity is defined as a change in distance over time, acceleration can be defined as the change in velocity over time. Acceleration can be calculated by taking the derivative of velocity with respect to time. If the velocity function was v(t), then the acceleration at time t would be a(t) = v'(t) = f"(t). A constant non-zero acceleration means that the velocity is increasing or decreasing linearly. If the acceleration is always 0 and velocity is non-zero, this means that velocity is constant and position is increasing or decreasing linearly.

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].

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

[math]\displaystyle{ \Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \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.

Here [1] is another glow script model that represents the change in velocity of the object with respect to time. It also represents how the velocity might affect the change in position of the object.

Velocity Time Graphs

Velocity-Time graphs describe the motion of the objects. Here, the time(s) is often represented at the x-axis since it is the independent variable, while the velocity is represented at the y-axis. The area under these graphs represents the displacement of the object while the slope represents the acceleration. Below is an example of a velocity time graph:

Velocity-Time Graph

Zero Velocity

When a vehicle has zero velocity, it is said to be stationary. Hence, it does not cover any distance per unit time. A graph with a solid line along the x-axis at y=0 is formed.

Constant Velocity

When a vehicle travels at a constant velocity in a particular direction, the acceleration is equal to zero. A graph for such motion consists of a flat line parallel of the x-axis. However, the line is not equal to y=0. In such situations, the vehicle is moving, but the velocity is constant at each time interval.

Increasing/Decreasing Velocity at Constant Rate

When a vehicle starts to travel faster/slower, but the change of velocity is constant per unit time, a graph with a constant positive/negative slope is obtained: Such graph also reflects constant acceleration/deceleration of the object.

Increasing Velocity at a Non-Constant Rate

When a vehicle starts to travel faster, and the change of velocity also increasing per unit time, a curved shaped graph pointing upwards is formed. Such graphs reflect increasing acceleration.

Decreasing Velocity at a Non-Constant Rate

When a vehicle starts to slow down and the change in velocity is also decreasing per unit time, the graph with a curve pointing downwards in obtained. This reflects that the object is decelerating.

Examples

Simple

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

[math]\displaystyle{ \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \frac{\lt -2, 8\gt - \lt 4,-1\gt }{3} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \frac{\lt -6, 9\gt }{3} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \lt -2, 3\gt }[/math]m/s

therefore

[math]\displaystyle{ \vec{v}(t=2) = \lt -2, 3\gt }[/math]m/s

Middling

A particle's position as a function of time is as follows: [math]\displaystyle{ \vec{r}(t) = \lt 6\sin(2 \pi t), -6t^3 + 10, e^{8t}\gt }[/math]

What is the particle's velocity as a function of time?

Solution:

[math]\displaystyle{ \vec{v} = \frac{d \vec{r}}{dt} }[/math]

[math]\displaystyle{ \vec{v} = \lt 12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}\gt }[/math]

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 [math]\displaystyle{ t\leq 3 }[/math],

[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 [math]\displaystyle{ t\gt 3 }[/math],

[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

Connectedness

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as Linear Momentum, Angular Momentum, and Magnetic Force depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

See Also

• Vectors
• Speed
• Speed vs Velocity
• Acceleration
• Kinematics
• Relative Velocity
• Derivation of Average Velocity

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. Sas, Wojciech. Angular Velocity Calculator. 02 Nov. 2022.

4. Beck, Kevin. Rotational Motion (Physics): What is it & Why it Matters. 28 Dec. 2020.