Speed
Kyleigh Quillian Fall 2025
This page defines and describes speed. Be sure to distinguish between speed and velocity. See Speed vs Velocity for similarities and differences.
Main Idea
Speed is a scalar quantity that describes how fast an object is moving. It is represented by the symbol [math]\displaystyle{ v }[/math], as opposed to [math]\displaystyle{ \vec{v} }[/math] and v, which denote velocity. Speed is defined as the magnitude of the rate of change of position with respect to time. In calculus terms, it is the time derivative of distance traveled. Speed is the magnitude of velocity. Speed is an instantaneous value, so it may change over time. Several properties in physics, such as kinetic energy, are functions of speed. Speed is given in unit distance per unit time. The SI unit for speed is the meter per second (m/s).
Why Speed Matters Physically
There is a lot more to speed than just "how fast something is moving" it is also used to help figure out
- how long a trip will end up taking
- the energy of a moving object
- the forces that are needed to stop or turn motion
- safety limits needed for vehicles or machines
- how motion appears to us in different reference frames
Instantaneous vs. Average Speed in Real Motion
Nothing on earth moves at a true constant speed. Even when something seems steady, there are little factors that affect this, such as:
- friction
- air resistance
- fluctuations of the engine output
- delays in reaction time
- changes in the road or terrain
All of these make it so that there is not a true constant speed. Because of this physicists really rely on instantaneous speed when they're describing motion at a specific moment, and average speed when looking at longer intervals or entire trips.
Speed in One Dimension vs. Multiple Dimensions
Speed behaves differently depending on the dimension it's in. In 1D, distance is just simply the magnitude of position change. But in 2D or 3D, distance accumulates along curved paths, so speed becomes especially useful because it does not depend on direction changes. This understanding is important to know when it comes to circular motion, orbital motion, and projectile motion.
A Mathematical Model
Instantaneous speed [math]\displaystyle{ v }[/math] is defined as:
[math]\displaystyle{ v = \frac{ds}{dt} }[/math]
where [math]\displaystyle{ s }[/math] is distance traveled and [math]\displaystyle{ t }[/math] is time.
Using Velocity to Define Speed:
Because speed is the magnitude of velocity
[math]\displaystyle{ v = |\vec{v}| }[/math]
It can be very useful to know this if you're given a position as a vector and need to figure out speed by differentiating.
Relativistic Speed
When speeds get close to the speed of light the normal mechanics don't work anymore and the speed becomes part of the relativistic factor:
[math]\displaystyle{ \gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}} }[/math]
This shows how speed affects length contraction along with time dilation.
Average Speed
Since speed is an instantaneous quantity, it can change over time. Over any given time interval, there is an average speed value denoted [math]\displaystyle{ v_{avg} }[/math]. The average speed over an interval of time [math]\displaystyle{ \Delta t }[/math] is given by
[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]
where [math]\displaystyle{ \Delta s }[/math] is the distance traveled during the time interval [math]\displaystyle{ \Delta t }[/math].
Average speed is often confused with average velocity. 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 speed over that time interval approaches the value of the instantaneous speed at any point in time within that interval.
If speed is constant over some time interval, the average speed over that time interval equals that constant speed.
Useful Real World Examples
- Speed Traps and Police Radar
Police radar guns are able to measure instantaneous speed, but not average speed. Tracking over time is how yo get average speed, which is why speed cameras on highways check:
[math]\displaystyle{ v_{\text{avg}} = \frac{\text{distance between checkpoints}}{\text{travel time}} }[/math]
- Running Track vs. GPS
Fitness trackers like Apple Watches, Fitbits, etc. tend to underestimate or overestimate average speed because they sample the position at discrete intervals instead of continuously. So the data collection is choppy not smooth.
Integral Relationships
Distance traveled is the time integral of speed:
[math]\displaystyle{ s(t) = \int v(t) \ dt }[/math].
Integrating the speed function as a definite integral from one time coordinate to another yields the distance traveled over that time interval:
[math]\displaystyle{ \Delta s_{t_1,t_2} = \int_{t_1}^{t_2} v(t) \ dt }[/math].
Speed from Acceleration
If acceleration is known:
[math]\displaystyle{ v(t) = \int a(t)\, dt + v_0 }[/math]
Distance from Acceleration
Combine twice:
[math]\displaystyle{ s(t) = \int \left( \int a(t)\, dt \right) dt }[/math]
This shows how speed connects acceleration and position.
Common Misconceptions
1. That speed can be negative:
This is incorrect. The velocity of an object can be negative but speed can't.
2. If displacement is zero then that means speed is zero:
Not always true. An object can move and also return to its start point so distance doesn't equal displacement.
3. If the instantaneous speed is zero at a point then the object stopped moving completely:
This is incorrect. For example the speed of a ball is zero for just a tiny second at its very highest point, but its overall average speed isn't.
4. The faster the object means the bigger the acceleration is:
Incorrect because speed and acceleration is not the same thing. Something can move really fast and have no acceleration.
Examples
Simple
1. A car starts at rest then accelerates so that its speed is [math]\displaystyle{ v(t)=4t }[/math]. Find the distance traveled in the first 5 seconds.
Solution:
[math]\displaystyle{ s = \int_0^5 4t\, dt = 4\left[\frac{t^2}{2}\right]_0^5 = 2(25) = 50 \text{ m} }[/math]
So the distance traveled is 50 m.
2. A runner runs at a circular track with radius 100m. The runner runs 3 laps in 250s at a constant speed. What is the speed of the runner?
Solution:
The runner moved at a constant speed, so that speed must also be the average speed the over the time interval 250s time interval over which the laps were completed. Let us therefore use the following formula:
[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]
[math]\displaystyle{ v_{avg} = \frac{2\pi*100}{250} }[/math]
[math]\displaystyle{ v_{avg} = 2.51 }[/math]m/s
The speed of the runner was 2.51m/s.
Middling
On graduation day, you excitedly throw your cap into the air! The cap leaves your hands when they are fully stretched upwards at a height of 2 meters above the ground. The cap then reaches a maximum height of 10 meters above the ground before falling straight back down to your hands, which have not moved since you threw the cap. If the cap was in the air for a total of four seconds, what was the average speed of the cap during that period? Your cap is light and therefore affected by air resistance, so do not assume that acceleration is constant.
Solution:
[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]
On its way up, the cap traveled from a height of 2m to a height of 10m, which is a distance of 8m. On its way down, the cap traveled from a height of 10m to a height of 2m, which is a distance of 8m. The total distance traveled by the cap in the 4 second time interval is therefore 8+8=16m.
[math]\displaystyle{ v_{avg} = \frac{16}{4} }[/math]
[math]\displaystyle{ v_{avg} = 4 }[/math]m/s.
Difficult
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:
Let us first find the velocity of the particle as a function of time. From there, the speed of the particle can be found by taking its magnitude.
[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]
[math]\displaystyle{ v = |\vec{v}| = \sqrt{(12 \pi \cos(2 \pi t))^2 + (-18t^2)^2 + (8e^{8t})^2} }[/math]
[math]\displaystyle{ v = \sqrt{144 \pi^2 \cos^2(2 \pi t) + 324t^4 + 64e^{16t}} }[/math]
Units and Conversions
Common units of speed:
- meters per second (m/s)
- kilometers per hour (km/h)
- miles per hour (mph)
Conversions that are useful:
1 m/s = 3.6 km/h
1 m/s [math]\displaystyle{ \equiv }[/math] 2.24 mph
Connectedness
Since the goal of the field of mechanics is to predict the motion of systems, speed is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as Kinetic Energy depend on the speeds of objects.
The use of speed 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 speeds of moving parts are important in virtually every type of machinery.
See Also
External links
The Physics Classroom: Speed and Velocity
References
1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.