# Speed and Velocity

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

## Speed and Velocity

Velocity ( $$v$$ ) (VEL) is the rate of change or displacement with time.  Velocity is a vector quantity having magnitude and direction. The scalar absolute value of magnitude of the velocity vector is the speed of the motion.

Speed ( $$u$$ ) is the rate of change or distance with time.  Speed is a scalar quantity having direction.

Speed is a scalar quantity having direction, some of these include area, density, energy, entropy, length, mass, power, pressure, temperature, volume, and work

Velocity is a vector quantity having magnitude and direction, some of these include acceleration, displacement, drag, force, lift, momentum, thrust, torque, and weight.

### Velocity Formula

$$v = \frac { d } { t }$$          $$velocity \;= \; \frac { displacement \; velocity } { time }$$

$$s = \frac { d } { t }$$          $$speed \;= \; \frac { distance \; speed } { time }$$

$$v = v_i + at$$          $$velocity \;=\; initial \; velocity \;+\; acceleration \;\;x\;\; time$$

$$v_i = v - at$$          $$initial \; velocity \;=\; velocity \;-\; acceleration \;\;x\;\; time$$

Where:

$$v$$ = velocity

$$a$$ = acceleration

$$d$$ = displacement velocity

$$d$$ = distance speed

$$s$$ = speed

$$t$$ = time

$$v_i$$ = initial velocity

Solve for:

$$t = \frac { d } { v }$$

$$t = \frac { d } { s }$$

## Angular Velocity

Angular velocity ( $$\theta$$ ) (angular speed) is how quickly an object moves through an angle.

### Angular Velocity Formula

$$\omega = \frac { \theta } { t }$$          $$angular \; velocity \;= \; \frac { angle \; distance } { time }$$

$$\omega = \frac { \Delta A } { t }$$          $$angular \; velocity \;=\; \frac { change \; in \; angle \; displacement } { time }$$

Where:

$$\omega$$ (Greek symbol omega) = angular velocity

$$\theta$$ (Greek symbol theta) = angle distance

$$\Delta A$$ = change in angle displacement

$$t$$ = time

## Average Velocity

Average velocity ($$\bar {v}$$ ) is the average of any given velocities where the acceleration is constant.

### Average Velocity FORMULA

$$\bar {v}= \frac { 1 } { 2 } \left( v_i \;-\; v_f \right)$$          $$average \;velocity \;= \; \frac { 1 } { 2 } \; \left( \; inital \; velocity \;-\; final \; velocity \; \right)$$

Where:

$$\bar {v}$$ = average velocity

$$v_f$$ = final velocity

$$v_i$$ = initial velocity

## Average Velocity change in Velocity

When an object make changes in its velocity at different times that is an average velocity of any given velocities.

### Average Velocity change in Velocity FORMULA

$$\bar {v}= \frac { v_1 \;+\; v_2 \;+\; v_3 ... v_n } { t_1 \;+\; t_2 \;+\; t_3 ... t_n }$$          $$average \;velocity \;= \; \frac { net \; total \; valocities } { total \;time \; of \; instances }$$

$$\bar {v}= \frac { v_1 \;+\; v_2 \;+\; v_3 ... v_n } { n }$$          $$average of velocity \;=\; \frac { velocity_1 \;+\; velocity_2 \;+\; velocity_3 \; ... \; velocity_n } { total \; time }$$

Where:

$$\bar {v}$$ = average of velocity

$$n$$ = total time

$$t$$ = time

$$v$$ = velocity

## Average Angular Velocity change in Velocity

When an object makes changes in its angular velocity at different times that is an average angular velocity of any given velocities.

### Average Angular Velocity change in Velocity FORMULA

$$\bar {\omega} \;= \; \frac { \omega_1 \;+\; \omega_2 \;+\; \omega_3 ... \omega_n } { t_1 \;+\; t_2 \;+\; t_3 ... t_n }$$          $$average \; angular \;velocity = \frac { net \; total \; angular \; valocities } { total \;time \; of \; instances }$$

$$\bar {\omega} = \frac { \omega_t } { t_t }$$          $$average \; angular \; velocities \;=\; \frac { total \; angular \; velocity } { total \; time }$$

Where:

$$\omega_t$$ (Greek symbol omega) = total angular velocity

$$t$$ = time

$$t_t$$ = total time

$$\omega$$ (Greek symbol omega) = angular velocity

$$\bar {\omega}$$ (Greek symbol omega) = average angular velocities

## Circular Velocity

Circular velocity ( $$v_c$$ ) is the velocity at which an object must move in order to keep a specific radius.

### Circular Velocity formula

$$v_c = \frac { 2 \pi r } { t }$$          $$circular \; velocity \;= \; \frac { 2 \;\;x\;\; Pi \;\;x\;\; radius } { time }$$

Where:

$$v_c$$ = circular velocity

$$\pi$$ = Pi

$$r$$ = radius

$$t$$ = time

Solve for:

$$r = \frac { v_c t } { 2 \pi }$$

$$t = \frac { 2 \pi r } { v_c }$$

## Escape Velocity

Escape velocity ( $$v_e$$ ) is the minimum velocity required to leave a planet or moon or the minimum velocity to overcome the pull of gravity.

### Escape Velocity formula

$$v_e = \sqrt { \frac { 2 G m} {r} }$$          $$escape \; velocity \;=\; \sqrt { \frac { \; 2 \;\;x\;\; universal \; gravitational \; constant \;\;x\;\; mass } { radius } }$$

Where:

$$v_e$$ = escape velocity

$$m$$ = mass of the plamet or moon

$$r$$ = radius from the center of mass (plamet or moon) to start point

## Final Velocity

### Final Velocity FORMULA

$$v_f = v_i \;+\; at$$          $$final \;velocity \;= \; initial \; velocity \;+\; \left( \;acceleration \;\;x\;\; time\; \right)$$

$$v_f = 2 \bar {v} \;-\; v_i$$          $$final velocity = 2 \;\;x\;\; average \; velocity \;-\; initial \; velocity$$

Where:

$$v_f$$ = final velocity

$$a$$ = acceleration

$$t$$ = time

$$\bar {v}$$ = average velocity

$$v_i$$ = initial velocity

## Fluid Velocity

Fluid velocity is how fast the process is traveling in a pipe.  To calculate the velocity of the fluid, use one of the equations below.

To get flow rate in Feet per Second

• Step 1 - Convert the flow rate to cubic feet per second.  This is done by multiplying the flow rate by one of the values shown on the Cubic Feet per Second page.
• Step 2 - divide by the flow area.  The information for flow area can be found on the Carbon Steel Pipe Properties page.  Make sure that the units match and convert the area into square feet.

### Fluid Velocity FORMULA

$$v = \frac { Q } { A }$$          $$velocity \;= \; \frac { volumetric \; flow \; rate } { area }$$

Where:

$$v$$ = velocity

$$A$$ = area

$$Q$$ = volumetric flow rate

## Impulse Velocity

Impulse is a change in momentum of an mass when a force is applied.

### Impulse Velocity formula

$$I = m \Delta v$$          $$impulse \; velocity \;= \; mass \;\;x\;\; difference \; in \; velocities$$

Where:

$$I$$ = impulse velocity

$$m$$ = mass

$$\Delta v$$ = velocity differential

Solve for:

$$m = \frac {I} {\Delta v}$$

$$\Delta v = \frac {I} {m}$$

## Initial Velocity

Initial velocity ( $$v_i$$ ) is the starting point at which motion begins.

### Initial Velocity Formula

$$v_i = v_f \;-\; at$$          $$initial \; velocity \;= \; final \; velocity \;-\; \left( \; acceleration \;\;x\;\; time \; taken \; \right)$$

$$v_i^2 = v_f^2 \;-\; 2as$$          $$initial \; velocity^2 \;=\; final \; velocity^2 \;-\; 2 \;\;x\;\; acceleration \;\;x\;\; displacement$$

$$v_i = \frac { s }{ t } \;-\; \frac { 1 }{ 2 } at$$          $$initial \; velocity \;=\; \frac { displacement }{ time \; taken } \;-\; \frac { 1 }{ 2 } \; acceleration \;\;x\;\; time \; taken$$

$$v_i = 2 \bar {v} \;-\; v_f$$          $$initial \; velocity \;=\; 2 \;\;x\;\; average \; velocity \;-\; final \; velocity$$

Where:

$$v_i$$ = initial velocity

$$a$$ = acceleration

$$s$$ = displacement

$$t$$ = time taken

$$\bar {v}$$ = average velocity

$$v_f$$ = final velocity

## Instantaneous Velocity

Instantaneous velocity is the derivative (a variable as it approaches 0) of distance with respect to time.

## Rate of Change in Velocity

The rate of change in velocity ( $$v_c$$ ) is the change in position or the ratio that shows the relationship of change of a fluid or object.

### Rate of Change in Velocity Formula

$$v_c = \frac {d}{t}$$          $$rate \; of \; change \; in \; velocity \;= \; \frac { displacement } { time }$$

$$v_c = \frac { v_f \;-\; v_i }{ t }$$          $$rate \; of \; change \; in \; velocity \;= \; \frac { final \; velocity \;-\; initial \; velocity } { time }$$

Where:

$$v_c$$ = rate of change in velocity

$$d$$ = displacement

$$t$$ = time taken tor change in velocity

$$v_f$$ = final velocity

$$v_i$$ = initial velocity

## Tangential Velocity

Tangential velocity ( $$v_t$$ ) is the velocity at any point tangent to a rotating object.

### Tangential Velocity formula

$$v_t = \frac { \omega } { r }$$          $$tangential \; velocity \;= \; \frac { angular \; velocity } { radius }$$

Where:

$$v_t$$ = tangential velocity

$$r$$ = radius

$$\omega$$ (Greek symbol omega) = angular velocity

## Terminal Velocity

Terminal velocity ( $$v_t$$ ) is when an object is falling under the influence of gravity but with no other influences.

### Terminal Velocity formula

$$v_t = \sqrt { \frac {2mg} {C_d \rho A} }$$          $$terminal \; velocity \;=\; \sqrt { \frac { \; 2 \;\;x\;\; mass \;\;x\;\; gravitational \; acceleration } { drag \; coefficient \;\;x\;\; fluid \; density} }$$

$$v_t = \sqrt { \frac {2W} {C_d \rho A} }$$          $$terminal \; velocity \;=\; \sqrt { \frac { \; 2 \;\;x\;\; weight } { drag \; coefficient \;\;x\;\; fluid \; density} }$$

Where:

$$v_t$$ = terminal velocity (maximum falling speed)

$$A$$ = area of the object

$$C_d$$ = drag coefficient

$$g$$ = gravitational acceleration

$$m$$ = mass

$$\rho$$ (Greek symbol rho) = fluid density

$$W$$ = weight of object

## Velocity Differential

Velocity differential is the average rate of change or displacement with time.

### Velocity Differential Formula

$$\Delta v = v_f \;- \; v_i$$          $$velocity \; differential \;= \; final \; velocity \;-\; initial \; velocity$$

Where:

$$\Delta v$$ = velocity differential

$$v_f$$ = final velocity

$$v_i$$ = initial velocity

Velocity gradient ( $$\nu$$ (Greek symbol nu) ) is how the velocity of a fluid changes between parallel planes or different points within the fluid.

$$\nu = \frac { v } { l }$$          $$velocity \; gradient \;=\; \frac { velocity } { distance }$$

Where:

$$\nu$$ (Greek symbol nu) = velocity gradient

$$l$$ = distance

$$v$$ = velocity