Velocity in Mechanics

Velocity describes how position changes with time. In one dimension it may be represented by a signed scalar component such as v_x. In two or three dimensions it is a vector: it has both magnitude and direction.

This article develops velocity from the position and displacement concepts of M01-01. The central ideas are

Δr- vavg = Δt

(1)

for average velocity and

v = dr- dt

(2)

for instantaneous velocity.

The magnitude of velocity is the speed,

v = |v|.

(3)

These definitions are simple, but they encode several important distinctions: average versus instantaneous quantities, vector velocity versus scalar speed, endpoint displacement versus path length, and coordinate components versus the physical velocity vector.

1 Average velocity

Suppose a particle moves from position r₁ at time t₁ to position r₂ at time t₂. The displacement is

Δr = r2 − r1,

(4)

and the elapsed time is

Δt = t2 − t1.

(5)

The average velocity over that interval is defined as

v = Δr-. avg Δt

(6)

Average velocity therefore points in the direction of the net displacement, not necessarily along every part of the path traveled.

In one-dimensional motion along the x axis,

x2 − x1 vx,avg = -------. t2 − t1

(7)

The SI unit is metre per second,

[v ] = m ∕s.

(8)

2 Instantaneous velocity as a limit

Average velocity describes an entire finite time interval. To determine the velocity at one instant, shrink the interval around that instant.

Let the particle have position r(t) at time t and r(t + Δt) a short time later. The displacement is

Δr = r(t + Δt ) − r(t).

(9)

The secant velocity is

Δr-. Δt

(10)

The instantaneous velocity is the limit

r(t + Δt ) − r(t) v(t) = lim ----------------. Δt→0 Δt

(11)

This is precisely the derivative of the position vector:

v (t) = dr. dt

(12)

Figure 1. The finite displacement Δr joins two nearby positions. As Δt shrinks, the secant velocity Δr∕Δt approaches the tangent velocity dr∕dt.

The direction of v is tangent to the trajectory at the particle’s instantaneous position, provided the derivative exists and the velocity is nonzero.

3 Velocity components in Cartesian coordinates

For a particle with

r(t) = x(t)ex + y(t)ey + z (t)ez,

(13)

where the Cartesian basis vectors are fixed in time,

dr- dx- dy- dz- v = dt = dtex + dt ey + dtez.

(14)

Thus

v = ˙x, v = y˙, v = ˙z. x y z

(15)

The dot notation means differentiation with respect to time.

Figure 2. The physical velocity vector may be resolved into Cartesian components. Its direction is tangent to the trajectory.

The speed is

∘ ------------ v = |v | = v2 + v2 + v2. x y z

(16)

4 Speed versus velocity

Velocity is a vector. Speed is the magnitude of that vector.

In one-dimensional motion, velocity can be positive, negative, or zero depending on direction. Speed is always nonnegative:

speed = |vx |.

(17)

A particle can therefore have the same speed at two instants while having different velocities because its direction has changed.

Likewise, average speed and the magnitude of average velocity are generally not equal.

Average speed is

average speed = total-distance-traveled-, Δt

(18)

whereas the magnitude of average velocity is

|Δr | |vavg| = -----. Δt

(19)

Because total distance traveled is at least as large as the displacement magnitude,

average speed ≥ |vavg|.

(20)

Equality occurs when the path length over the interval equals the straight-line endpoint separation.

Figure 3. Average speed uses total path length, while average velocity uses the endpoint displacement. Reversal or curvature can make the two differ greatly.

5 Velocity from a position-time graph

For one-dimensional motion, a graph of x versus t contains velocity information in its slope.

Over a finite interval,

v = Δx-, x,avg Δt

(21)

which is the slope of a secant line through two points on the position-time graph.

At an instant,

dx vx = --, dt

(22)

which is the slope of the tangent line.

Figure 4. Average velocity is represented by a secant slope on an x-t graph; instantaneous velocity is the tangent slope.

A positive slope means positive velocity, a negative slope means negative velocity, and a horizontal tangent means zero instantaneous velocity.

A steeper position-time slope corresponds to a larger speed in one dimension.

6 Velocity along a curved trajectory

For motion along a smooth curve, the infinitesimal displacement may be written

dr = T ds,

(23)

where T is the unit tangent vector and ds is a positive differential of path length.

Dividing by dt gives

v = dr-= T ds-. dt dt

(24)

Therefore

v = vT,

(25)

with

ds v = ---. dt

(26)

This gives a precise geometric interpretation: speed tells how rapidly path length is accumulated, while the unit tangent gives the instantaneous direction of motion.

Consequently, the distance traveled from t₁ to t₂ is

∫ t2 s = v(t)dt. 12 t1

(27)

This integral uses speed, not signed one-dimensional velocity and not the vector velocity itself.

7 Velocity depends on the reference frame

Velocity is not an absolute quantity in Newtonian mechanics. If frame S′ moves with constant velocity V relative to frame S, then the Galilean velocity transformation is

v′ = v − V.

(28)

Thus two inertial observers can assign different velocities to the same particle.

This frame dependence does not make velocity ambiguous. It means that every velocity statement must be understood relative to a specified observer or reference frame.

M00-06 develops these transformations in detail.

8 Worked example 1: average velocity and average speed

A student walks 30 m east in 20 s and then 10 m west in 10 s. Find the average velocity and average speed for the full trip.

Take east as positive. The total displacement is

Δx = 30 − 10 = 20 m.

(29)

The total elapsed time is

Δt = 30 s.

(30)

Therefore

20- vavg = 30 = 0.667 m∕s

(31)

in the eastward direction.

The distance traveled is

d = 30 + 10 = 40 m.

(32)

Hence the average speed is

40 ---= 1.33m ∕s. 30

(33)

The two averages differ because part of the path retraces earlier motion.

9 Worked example 2: instantaneous velocity from position

A particle moves along the x axis according to

x(t) = 2t3 − 9t2 + 12t,

(34)

with x in metres and t in seconds. Find the velocity at t = 2 s and determine whether the particle is moving in the positive or negative x direction.

Differentiate:

v (t) = dx-= 6t2 − 18t + 12. x dt

(35)

At t = 2 s,

vx(2) = 24 − 36 + 12 = 0.

(36)

Thus the particle is instantaneously at rest at that instant. The sign of velocity just before and just after t = 2 s must be checked to determine how the motion changes through that point.

10 Worked example 3: velocity of a particle in two dimensions

Let

r(t) = (3t2)ex + (4t − t2)ey,

(37)

with position in metres. Find the velocity and speed at t = 1 s.

Differentiate componentwise:

v (t) = 6t e + (4 − 2t)e . x y

(38)

At t = 1 s,

v(1) = 6ex + 2ey m ∕s.

(39)

The speed is

√ -2----2 √--- v = 6 + 2 = 40 ≈ 6.32 m ∕s.

(40)

The velocity vector gives both the instantaneous speed and the tangent direction of motion.

11 Common mistakes

Confusing average speed with the magnitude of average velocity.
Using total distance traveled in the average-velocity formula instead of displacement.
Treating instantaneous velocity as x∕t rather than dx∕dt.
Forgetting that negative one-dimensional velocity indicates direction, not negative speed.
Reading the height of an x-t graph as velocity. Velocity is represented by the graph’s slope.
Assuming zero instantaneous velocity means zero acceleration. A turning point can have v = 0 while acceleration is nonzero.
Forgetting to specify the reference frame when comparing velocities measured by different observers.

12 Practice problems

M01-02-P01

A particle moves from x = −5 m at t = 2 s to x = 11 m at t = 6 s. Find the average velocity.

M01-02-P02

A runner completes one 400 m lap in 80 s and stops at the starting point. Find the average speed and average velocity.

M01-02-P03

For x(t) = 5 + 3t − 2t², find v_x(t) and evaluate it at t = 2 s.

M01-02-P04

For x(t) = t³ − 6t² + 9t, find every time at which the particle is instantaneously at rest.

M01-02-P05

A particle has r(t) = 2te_x + t²e_y. Find v(t) and its speed at t = 3 s.

M01-02-P06

A CAR travels 60 km east in 1.0 h and then 30 km west in 0.5 h. Find the average velocity and average speed for the whole trip.

M01-02-P07

A particle moves on a circle of radius R at constant speed v. Explain why the velocity is not constant even though the speed is constant.

M01-02-P08

Suppose v = 3e_x − 4e_y m∕s. Find the speed and the direction angle measured from +x.

M01-02-P09

A particle moves with v_x(t) = 4 − 2t and begins at x(0) = 3 m. Find x(t) and the displacement from t = 0 to t = 3 s.

M01-02-P10

A frame S′ moves at 5 m∕s in the +x direction relative to frame S. A particle has velocity 12 m∕s in S. Find its velocity in S′.

13 Answer check

P01: 4.0 m∕s.
P02: average speed 5.0 m∕s; average velocity 0.
P03: v_x = 3 − 4t; v_x(2) = −5 m∕s.
P04: t = 1 s and t = 3 s.
P05: v = 2e_x + 2te_y; at t = 3, v = ≈ 6.32 m∕s.
P06: average velocity 20 km∕h east; average speed 60 km∕h.
P07: the tangent direction changes continuously around the circle.
P08: speed 5 m∕s; direction 53.1^∘ below the +x axis.
P09: x(t) = 3 + 4t − t²; Δx = 3 m from 0 to 3 s.
P10: 7 m∕s in the +x direction.

14 Bridge to acceleration

Velocity describes the first time derivative of position. The next kinematic quantity measures how velocity changes:

dv- d2r- a = dt = dt2.

(41)

M01-03 develops acceleration, including the crucial fact that acceleration can arise from a change in speed, a change in direction, or both.

15 Source and licensing note

This article upgrades and expands the existing PhysicsLibrary entry Velocity (object id 74). The treatment was also cross-checked against the archived 2016 University Physics Volume 1 material and the UCD Physics 9A classical-mechanics sequence. The organization, derivations, examples, exercises, and figures in this M01-02 entry were prepared specifically for PhysicsLibrary. Unless otherwise noted, the intended license is CC BY-SA 4.0.

References

[1] PhysicsLibrary, Velocity, object id 74.

[2] PhysicsLibrary, M01-01: Position and Displacement in Mechanics.

[3] OpenStax / cnxuniphysics, University Physics Volume 1, archived 2016 BCcampus clone, CC BY 4.0.

[4] University of California, Davis, Physics 9A: Classical Mechanics, Physics LibreTexts, CC BY-SA 4.0.

[5] J. R. Taylor, Classical Mechanics, University Science Books, 2005. Reference used for scope and mechanics notation; no restricted text reproduced.

"Velocity in Mechanics" is owned by bloftin.
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