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:

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.

The average velocity is the same as the velocity averaged over time – that is to say, its time-weighted average, which may be calculated as the time integral of the velocity:

$${\displaystyle {\boldsymbol {\bar {v}}}={1 \over t_{1}-t_{0}}\int _{t_{0}}^{t_{1}}{\boldsymbol {v}}(t)\ dt,}$$

where we may identify

$${\displaystyle \Delta {\boldsymbol {x}}=\int _{t_{0}}^{t_{1}}{\boldsymbol {v}}(t)\ dt}$$

and

$${\displaystyle \Delta t=t_{1}-t_{0}.}$$

• When a particle moves with different uniform speeds v₁, v₂, v₃, ..., v_n in different time intervals t₁, t₂, t₃, ..., t_n respectively, then average speed over the total time of journey is given as
  $${\displaystyle {\boldsymbol {\bar {v}}}={v_{1}t_{1}+v_{2}t_{2}+v_{3}t_{3}+\dots +v_{n}t_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}}$$

  

If t₁ = t₂ = t₃ = ... = t, then average speed is given by the arithmetic mean of the speeds

$${\displaystyle {\boldsymbol {\bar {v}}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}}$$

• When a particle moves different distances s₁, s₂, s₃,..., s_n with speeds v₁, v₂, v₃,..., v_n respectively, then the average speed of the particle over the total distance is given as

$${\displaystyle {\boldsymbol {\bar {v}}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}}$$

If s₁ = s₂ = s₃ = ... = s, then average speed is given by the harmonic mean of the speeds

$${\displaystyle {\boldsymbol {\bar {v}}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}}$$

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.)

If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time:

$${\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {x}}}{\Delta t}}={\frac {d{\boldsymbol {x}}}{dt}}.}$$

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, x. In calculus terms, the integral of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds to the yellow area under the curve labeled s (s being an alternative notation for displacement).

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). 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.

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, acceleration is defined as the derivative of velocity with respect to time:

From there, we can obtain an expression for velocity as the area under an a(t) acceleration vs. time graph. As above, this is done using the concept of the integral:

$${\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.}$$

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, it is trivial to show that

$${\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t}$$

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:

$${\displaystyle v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}}$$

$${\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}}$$

$${\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})}$$

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.

The kinetic energy of a moving object is dependent on its velocity and is given by the equation

ignoring special relativity, where E_k is the kinetic energy and m is the mass. Kinetic energy is a scalar quantity as it depends on the square of the velocity, however a related quantity, momentum, is a vector and defined by

In special relativity, the dimensionless Lorentz factor appears frequently, and is given by

where γ is the Lorentz factor and c is the speed of light.

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.

In the one-dimensional case,[3] 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.