Velocity
In our everyday lives, we often use 'speed' and 'velocity' interchangeably, but in the world of physics, these terms hold distinct and crucial meanings. Velocity isn't just about how fast something is moving—it's also about where it's going. This subtle yet profound difference underpins our understanding of everything from a car rounding a bend to a spacecraft escaping Earth's gravity, revealing a richer, more directional view of motion. Velocity is a vector quantity, meaning it includes both the speed and the specific direction of an object's motion. Understanding instantaneous velocity requires calculus, as it represents the exact rate of position change at a single moment in time. Many fundamental physical phenomena, from momentum and kinetic energy to drag and orbital mechanics, are deeply intertwined with an object's velocity.
AI Summary
In our everyday lives, we often use 'speed' and 'velocity' interchangeably, but in the world of physics, these terms hold distinct and crucial meanings. Velocity isn't just about how fast something is moving—it's also about where it's going. This subtle yet profound difference underpins our understanding of everything from a car rounding a bend to a spacecraft escaping Earth's gravity, revealing a richer, more directional view of motion.
- Velocity is a vector quantity, meaning it includes both the speed and the specific direction of an object's motion.
- Understanding instantaneous velocity requires calculus, as it represents the exact rate of position change at a single moment in time.
- Many fundamental physical phenomena, from momentum and kinetic energy to drag and orbital mechanics, are deeply intertwined with an object's velocity.
Defining Velocity
At its heart, velocity describes how quickly an object changes its position, but with a critical addition—it tells us the direction of that change. Think of it as a complete picture of motion, whereas speed is just one piece of that picture.
Speed vs. Velocity
This is where the scientific distinction becomes vital. Speed is a scalar quantity, only telling us 'how fast.' Velocity, however, is a vector quantity, encompassing both 'how fast' and 'in what direction.' To have constant velocity, an object must maintain both a steady speed AND a consistent direction.
Consider a car driving at a steady 20 kilometers per hour around a circular track. Its speed is constant. But its velocity is continuously changing, because its direction is constantly shifting. This change in velocity—even if speed remains the same—means the car is accelerating!
Average Velocity
When we look at motion over a longer period, we can talk about average velocity. This is simply the total change in an object's position, or its displacement, divided by the time it took for that change to occur. It doesn't tell us what happened in between, just the overall effect.
Here, $\Delta \boldsymbol{s}$ represents the displacement (change in position, a vector), and $\Delta t$ is the duration of the time period. Notice the bar over the 'v' and the bold 's' and 'v' — these often signify vector quantities and averages.
\bar{\boldsymbol{v}} = \frac{\Delta \boldsymbol{s}}{\Delta t}Instantaneous Velocity
What if we want to know the velocity at a precise moment, not just an average over a duration? That's where instantaneous velocity comes in. It's the velocity an object has at any single point in time, essentially if it stopped accelerating right then.
Mathematically, instantaneous velocity is found using calculus, specifically as the derivative of an object's position with respect to time. This tells us the exact rate of change of position at that specific instant.
This equation shows that as the time interval $\Delta t$ shrinks towards zero, the average velocity approaches the instantaneous velocity $\boldsymbol{v}$. In a position-time graph, this would be the slope of the tangent line at any given point.
\boldsymbol{v}=\lim _{{\Delta t}\to 0}{\frac{\Delta \boldsymbol{s}}{\Delta t}}={\frac{d\boldsymbol{s}}{dt}}.Units of Velocity
Because velocity is a change in position (measured in meters) over a change in time (measured in seconds), its standard unit in the International System of Units (SI) is meters per second (m/s). So, 5 m/s east is a velocity, while just 5 m/s is a speed.
Velocity's Kinematic Connections
Relationship to Acceleration
Just as velocity is the rate of change of position, acceleration is the rate of change of velocity. If an object's velocity changes—either its speed, its direction, or both—it is accelerating. This relationship is also defined through derivatives.
Here, $\boldsymbol{a}$ represents the instantaneous acceleration. Conversely, if you know the acceleration of an object over time, you can integrate it to find the object's velocity. Think of it as finding the 'area under the curve' of an acceleration-time graph.
\boldsymbol{a}={\frac{d\boldsymbol{v}}{dt}}.In the special case of constant acceleration, simpler 'suvat' equations, like $\boldsymbol{v} = \boldsymbol{u} + \boldsymbol{a}t$, can directly relate initial velocity ($\boldsymbol{u}$), final velocity ($\boldsymbol{v}$), acceleration ($\boldsymbol{a}$), and time ($t$). These are cornerstones of classical mechanics.
\boldsymbol{v}=\int \boldsymbol{a}\,dt.Quantities Dependent on Velocity
Momentum
One of the most fundamental concepts tied to velocity is momentum. It's a measure of the 'quantity of motion' an object possesses. In classical mechanics, an object's momentum is directly proportional to its mass and its velocity.
As you can see, momentum ($\boldsymbol{p}$) is a vector quantity, inheriting its direction from velocity ($\boldsymbol{v}$). A heavier object or a faster object will have more momentum.
\boldsymbol{p}=m\boldsymbol{v}Kinetic Energy
Another crucial concept is kinetic energy, the energy an object possesses due to its motion. This energy doesn't depend on direction, only on speed, making it a scalar quantity. It increases dramatically as velocity increases.
Notice the velocity term $v$ is squared. This means doubling an object's speed quadruples its kinetic energy—a principle with significant implications, from car crashes to bullet impacts.
E_{\text{k}}={\tfrac {1}{2}}mv^{2}Drag (Fluid Resistance)
When an object moves through a fluid, like air or water, it experiences a resistance force called drag. This force always opposes the object's motion and, for many real-world scenarios, is heavily dependent on the object's velocity.
This equation shows that drag force ($FD$) is proportional to the square of the object's speed ($v$). Other factors include the fluid's density ($\rho$), the object's cross-sectional area ($A$), and its drag coefficient ($CD$), which accounts for its shape.
F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,AEscape Velocity
Imagine launching a rocket from Earth. To truly break free from our planet's gravitational pull and never fall back, it needs to reach a specific speed known as escape velocity. It's the minimum speed required to escape the gravitational field of a massive body.
This formula shows that escape velocity ($ve$) depends on the mass of the planet ($M$), the gravitational constant ($G$), and the distance from the planet's center ($r$). For Earth's surface, this is about 11,200 m/s (over 25,000 mph!), regardless of the launch direction.
v_{\text{e}}={\sqrt {\frac {2GM}{r}}}The Lorentz Factor (Special Relativity)
When objects approach the speed of light, classical physics breaks down, and special relativity takes over. Here, velocity plays a role in the Lorentz factor, a dimensionless quantity that describes how measurements of time, length, and mass are affected by relative motion.
As an object's speed ($v$) approaches the speed of light ($c$), the Lorentz factor ($\gamma$) increases dramatically. This factor is crucial for understanding phenomena like time dilation and length contraction at relativistic speeds.
\gamma ={\frac{1}{\sqrt {1-{\frac{v^{2}}{c^{2}}}}}}Relative Velocity
Motion is always observed relative to something else. Relative velocity describes the velocity of an object as seen from the perspective of another moving object or a specific reference frame. It's how we compare the motion of multiple things.
If object A moves with velocity $\boldsymbol{v}$ and object B moves with velocity $\boldsymbol{w}$, both relative to a stationary observer, then the velocity of A relative to B is simply the difference between their individual velocity vectors.
This vector subtraction helps us understand scenarios like two cars passing each other, or how a ball thrown on a moving train appears from outside the train. In Newtonian mechanics, this relative velocity is the same regardless of the chosen stationary observer.
\boldsymbol{v}_{A\text{ relative to }B}=\boldsymbol{v}-\boldsymbol{w}Velocity in Different Coordinate Systems
Cartesian Coordinates
To precisely describe velocity in 2D or 3D space, we often break it down into components using a coordinate system. In Cartesian coordinates (x, y, z), velocity is described by its movement along each axis.
A two-dimensional velocity vector can be expressed as $\boldsymbol{v} = \langle vx, vy \rangle$, where $vx$ is the velocity along the x-axis and $vy$ is the velocity along the y-axis. Its overall speed (magnitude) is found using the Pythagorean theorem.
For three dimensions, we simply add a z-component, making the vector $\boldsymbol{v} = \langle vx, vy, vz \rangle$. The magnitude formula extends naturally.
|\boldsymbol{v}| = \sqrt{v_{x}^{2}+v_{y}^{2}}.These components allow us to analyze complex motions by breaking them into simpler, independent parts.
|\boldsymbol{v}| = \sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}.Polar Coordinates
Another powerful way to describe 2D motion, especially for objects moving around a central point, is through polar coordinates. Here, velocity is described by two components: radial and transverse velocity.
Article
Velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector quantity, meaning that both magnitude and direction are needed to define it (velocity vector). The scalar absolute value (magnitude) of velocity is called speed, a quantity that is measured in metres per second (m/s or m⋅s−1) in the SI (International System of Units) system. For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration.
Definition
Average velocity
The average velocity of an object over a period of time is its change in position, ${\displaystyle \Delta s}$, divided by the duration of the period, ${\displaystyle \Delta t}$, given mathematically as ${\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.}$
Instantaneous velocity
Example of a velocity vs. time graph, and the relationship between velocity v on the y-axis, acceleration a (the three green tangent lines represent the values for acceleration at different points along the curve) and displacement s (the yellow area under the curve.)
The instantaneous velocity of an object is the limit average velocity as the time interval approaches zero. At any particular time t, it can be calculated as the derivative of the position with respect to time: ${\displaystyle {\boldsymbol {v}}=\lim {{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.}$
From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, s. In calculus terms, the integral of the velocity function v(t) is the displacement function s(t). In the figure, this corresponds to the yellow area under the curve. ${\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.}$
Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment.
Difference between speed and velocity
Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
While the terms speed and velocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction.
To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed.
For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.
Units
Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s).
Equation of motion
Average velocity
Velocity is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v(t), over some time period Δt. Average velocity can be calculated as: ${\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {x} }{\Delta t}}={\frac {\int {t{0}}^{t{1}}\mathbf {v} (t)\,dt}{t{1}-t{0}}}.}$
The average velocity is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction.
In terms of a displacement-time (x vs. t) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity.
Special cases
• When a particle moves with different uniform speeds v1, v2, v3, ..., vn in different time intervals t1, t2, t3, ..., tn respectively, then average speed over the total time of journey is given as ${\displaystyle {\bar {v}}={\frac {v{1}t{1}+v{2}t{2}+v{3}t{3}+\dots +v{n}t{n}}{t{1}+t{2}+t{3}+\dots +t{n}}}}$ If t1 = t2 = t3 = ... = t, then average speed is given by the arithmetic mean of the speeds ${\displaystyle {\bar {v}}={\frac {v{1}+v{2}+v{3}+\dots +v{n}}{n}}={\frac {1}{n}}\sum {i=1}^{n}{v{i}}}$ • When a particle moves different distances s1, s2, s3,..., sn with speeds v1, v2, v3,..., vn respectively, then the average speed of the particle over the total distance is given as ${\displaystyle {\bar {v}}={s{1}+s{2}+s{3}+\dots +s{n} \over t{1}+t{2}+t{3}+\dots +t{n}}={{s{1}+s{2}+s{3}+\dots +s{n}} \over {{s{1} \over v{1}}+{s{2} \over v{2}}+{s{3} \over v{3}}+\dots +{s{n} \over v{n}}}}}$If s1 = s2 = s3 = ... = s, then average speed is given by the harmonic mean of the speeds ${\displaystyle {\bar {v}}=n\left({1 \over v{1}}+{1 \over v{2}}+{1 \over v{3}}+\dots +{1 \over v{n}}\right)^{-1}=n\left(\sum {i=1}^{n}{\frac {1}{v{i}}}\right)^{-1}.}$
Relationship to acceleration
Although velocity is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v(t) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time: ${\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.}$
From there, velocity is expressed as the area under an a(t) acceleration vs. time graph. As above, this is done using the concept of the integral:
${\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.}$
Constant acceleration
In the special case of constant acceleration, velocity can be studied using the suvat equations. By considering a as being equal to some arbitrary constant vector, this shows ${\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t}$ with v as the velocity at time t and u as the velocity at time t = 0. By combining this equation with the suvat equation x = ut + at2/2, it is possible to relate the displacement and the average velocity by ${\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.}$ It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows: ${\displaystyle {\begin{aligned}v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}&=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)\\&=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}\end{aligned}}}$ ${\displaystyle {\begin{aligned}(2{\boldsymbol {a}})\cdot {\boldsymbol {x}}&=(2{\boldsymbol {a}})\cdot \left({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2}\right)\\[1ex]&=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}\end{aligned}}}$ ${\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})}$ where v = |v| etc.
The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated.
Quantities that are dependent on velocity
Momentum
In classical mechanics, Newton's second law defines momentum, p, as a vector that is the product of an object's mass and velocity, given mathematically as${\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}}$where m is the mass of the object.
Kinetic energy
The kinetic energy of a moving object is dependent on its velocity and is given by the equation${\displaystyle E{\text{k}}={\tfrac {1}{2}}mv^{2}}$where Ek is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity.
Drag (fluid resistance)
In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, ${\displaystyle F{D}}$, is dependent on the square of velocity and is given as${\displaystyle F{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C{D}\,A}$where
• ${\displaystyle \rho }$ is the density of the fluid, • ${\displaystyle v}$ is the speed of the object relative to the fluid, • ${\displaystyle A}$ is the cross sectional area, and • ${\displaystyle C{D}}$ is the drag coefficient – a dimensionless number.
Escape velocity
Escape velocity is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is${\displaystyle v{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},}$where G is the gravitational constant and g is the gravitational acceleration. The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path.
The Lorentz factor of special relativity
In special relativity, the dimensionless Lorentz factor appears frequently, and is given by${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$where γ is the Lorentz factor and c is the speed of light.
Relative velocity
Velocity
Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles.
Consider an object A moving with velocity vector v and an object B with velocity vector w; these absolute velocities are typically expressed in the same inertial reference frame. Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: ${\displaystyle {\boldsymbol {v}}{A{\text{ relative to }}B}={\boldsymbol {v}}-{\boldsymbol {w}}}$ Similarly, the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: ${\displaystyle {\boldsymbol {v}}{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}}$ Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest.
In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.
Scalar velocities
In the one-dimensional case, the velocities are scalars and the equation is either: ${\displaystyle v{\text{rel}}=v-(-w),}$ if the two objects are moving in opposite directions, or: ${\displaystyle v{\text{rel}}=v-(+w),}$ if the two objects are moving in the same direction.
Coordinate systems
Cartesian coordinates
In multi-dimensional Cartesian coordinate systems, velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as
The two-dimensional velocity vector is then defined as ${\displaystyle {\textbf {v}}=\langle v{x},v{y}\rangle }$. The magnitude of this vector represents speed and is found by the distance formula as
${\displaystyle |\mathbf {v} |={\sqrt {v{x}^{2}+v{y}^{2}}}.}$
In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as
The three-dimensional velocity vector is defined as ${\displaystyle {\textbf {v}}=\langle v{x},v{y},v{z}\rangle }$ with its magnitude also representing speed and being determined by
${\displaystyle |\mathbf {v} |={\sqrt {v{x}^{2}+v{y}^{2}+v{z}^{2}}}.}$
While some textbooks use subscript notation to define Cartesian components of velocity, others use ${\displaystyle u}$, ${\displaystyle v}$, and ${\displaystyle w}$ for the ${\displaystyle x}$-, ${\displaystyle y}$-, and ${\displaystyle z}$-axes respectively.
Polar coordinates
Representation of radial and tangential components of velocity at different moments of linear motion with constant velocity of the object around an observer O (it corresponds, for example, to the passage of a car on a straight street around a pedestrian standing on the sidewalk). The radial component can be observed due to the Doppler effect, the tangential component causes visible changes of the position of the object.
In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin, and a transverse velocity, perpendicular to the radial one. Both arise from angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system).
The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin. ${\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}{T}+{\boldsymbol {v}}{R}}$ where
• ${\displaystyle {\boldsymbol {v}}{T}}$ is the transverse velocity • ${\displaystyle {\boldsymbol {v}}{R}}$ is the radial velocity.
The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. ${\displaystyle v{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}}$ where ${\displaystyle {\boldsymbol {r}}}$ is position and ${\displaystyle {\hat {\boldsymbol {r}}}}$ is the radial direction.
The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed ${\displaystyle \omega }$ and the radius (the magnitude of the position). ${\displaystyle v{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|}$ such that ${\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.}$
Angular momentum in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. ${\displaystyle L=mrv{T}=mr^{2}\omega }$ where
• ${\displaystyle m}$ is mass • ${\displaystyle r=|{\boldsymbol {r}}|.}$
The expression ${\displaystyle mr^{2}}$ is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.