Relative Motion

Motion is always described relative to a chosen observer or reference frame. A passenger can move forward relative to a train while the train moves forward relative to the ground; a boat can move north relative to the water while the water itself flows east relative to the riverbank. Relative motion kinematics provides a systematic way to combine these descriptions.

The central idea is vector addition. If object P is observed from frame B, and frame B moves relative to frame G, then the velocity of P relative to G is

(1)

The notation P∕G means “P relative to G.” Keeping the order of the labels explicit is one of the best ways to avoid sign errors.

1 Relative position

Let O_G be the origin of frame G, let O_B be the origin of a second frame B, and let P be a particle. Define

(2)

(3)

and

(4)

The position vectors form a head-to-tail triangle:

(5)

This relation is purely geometric. The corresponding velocity and acceleration equations follow by differentiation.

2 Relative velocity for translating frames

Differentiate the relative position equation with respect to time. If the axes of B remain parallel to the axes of G, so that the two frames differ only by translation, then

(6)

Therefore

(7)

Because vector addition is commutative, this is often written in the visually convenient order

(8)

The quantities must still refer to a consistent chain of frames. For example,

(9)

3 Relative acceleration

Differentiating again gives

(10)

If frame B moves with constant velocity relative to G, then

(11)

so

(12)

This is the Newtonian acceleration invariance used in Galilean inertial frames. It does not hold for an accelerating origin, and additional terms appear if the axes rotate.

4 One-dimensional relative motion

In one dimension, the vector equation reduces to signed scalar addition. Suppose the positive x direction is east. Then

(13)

The signs carry the directional information. A train moving west has negative velocity if east is chosen positive.

For two objects A and B measured in the same frame G, the velocity of A relative to B is

(14)

In one dimension,

(15)

If both objects move east at 28 m/s and 20 m/s, then object A moves east relative to B at 8 m/s. If B instead moves west at 20 m/s, then v_B = −20 m/s and

(16)

The familiar “closing speed” in a straight line approach is therefore just a relative velocity calculation with a sign convention.

5 Two-dimensional relative motion

In two dimensions, the relative motion relation must be applied component by component:

(17)

A classic example is a boat crossing a river. The boat has a velocity relative to the water, the water has a velocity relative to the ground, and the ground observer sees their vector sum.

If

(18)

and the current is

(19)

then

(20)

The magnitude is

(21)

and the direction follows from the component ratio.

6 Aiming to cancel a cross flow

Sometimes the desired ground track is specified instead of the heading relative to the moving medium. In that case, the unknown relative velocity must be chosen so that one component of the vector sum has the required value.

For example, if an aircraft must travel due north while a wind blows east, the aircraft must point somewhat west of north so that the westward air relative component cancels the eastward wind component.

If the airspeed is v_a, the eastward wind speed is v_w, and the desired east-west ground component is zero, then

(22)

Therefore

(23)

If the aircraft’s airspeed magnitude is fixed,

(24)

so

(25)

This is the same vector addition problem as the river crossing example, but solved for a different unknown.

7 Galilean transformation

Let frame B move at constant velocity V relative to frame G. If their origins coincide at t = 0, then

(26)

From the relative position equation,

(27)

Using the conventional notation

(28)

we obtain the Galilean position transformation. Differentiation gives

(29)

and, for constant V,

(30)

In Newtonian mechanics the time coordinate is also taken to be common:

(31)

At speeds comparable with the speed of light, Galilean transformations must be replaced by Lorentz transformations; that subject belongs to special relativity rather than classical kinematics.

8 Relative velocity between two particles

Suppose particles A and B have velocities v_A and v_B in the same inertial frame. The velocity of A relative to B is

(32)

Likewise,

(33)

Thus the two relative velocities have equal magnitudes and opposite directions.

The magnitude

(34)

is the instantaneous rate at which their separation vector changes in magnitude only when the relative velocity happens to lie along the line joining them. In general, relative velocity changes both the separation magnitude and its direction.

9 Worked example 1: walking inside a moving train

A train moves east at

(35)

A passenger walks east inside the train at

(36)

The passenger’s ground velocity is

(37)

Therefore

(38)

If the passenger instead walks west at 1.50 m/s relative to the train, then v_P∕T = −1.50 m/s and the ground speed is 18.5 m/s east.

10 Worked example 2: relative speed of two cars

Car A travels east at

(39)

while car B travels east at

(40)

The velocity of A relative to B is

(41)

Thus an observer in car B sees car A move east at

(42)

If car B reverses direction and travels west at 20 m/s, then v_B = −20 m/s and

(43)

The same subtraction rule handles both cases; only the signed velocity changes.

11 Worked example 3: boat crossing a river

A river is 120 m wide. A boat is pointed straight north and moves relative to the water at

(44)

The river current is

(45)

The boat’s ground velocity is

(46)

Its ground speed is

(47)

The northward component determines the crossing time:

(48)

During that time the eastward current produces a drift

(49)

The ground track direction is

(50)

so

(51)

12 Worked example 4: aircraft correcting for a crosswind

An aircraft has airspeed

(52)

A wind blows due east at

(53)

The pilot wants the ground track to be due north. The aircraft must therefore have a westward air-relative component of 50 km/h:

(54)

The northward component is

(55)

The required heading angle west of north satisfies

(56)

Thus

(57)

Because the east-west components cancel, the ground speed is

(58)

13 Limits of the simple addition law

The formula

(59)

in the form used here assumes that the two coordinate bases remain parallel. If frame B rotates relative to G, differentiating a vector expressed in the rotating basis produces additional terms involving the angular velocity of the frame. Those terms lead to Coriolis, centrifugal, and Euler contributions and are treated later in non inertial frame mechanics.

The Galilean law is also a low speed approximation. At relativistic speeds, velocity addition is governed by special relativity.

14 Practice problems

1. A train moves east at 18 m/s. A passenger walks west at 2.0 m/s relative to the train. Find the passenger’s velocity relative to the ground.
2. Car A moves east at 25 m/s and car B moves east at 17 m/s. Find v_A∕B and v_B∕A.
3. Car A moves east at 22 m/s while car B moves west at 15 m/s. Find the magnitude and direction of v_A∕B.
4. A boat moves north at 5.0 m/s relative to the water while a current flows east at 2.0 m/s. Find the boat’s ground velocity magnitude and direction east of north.
5. The river in Problem 4 is 150 m wide. If the boat is pointed straight north, find the crossing time and downstream drift.
6. An aircraft has airspeed 250 km/h and encounters a 60 km/h wind blowing east. What heading west of north produces a due north ground track? What is the resulting ground speed?
7. Frame B moves at constant velocity 8e_x m/s relative to frame G. A particle has velocity (3e_x + 4e_y) m/s in B. Find its velocity in G.
8. A particle has acceleration (2e_x − 3e_y) m/s² in one inertial frame. What acceleration is measured in another frame moving at constant velocity relative to the first?
9. Two particles have velocities v_A = (6e_x + 2e_y) m/s and v_B = (1e_x + 5e_y) m/s. Find v_A∕B and its magnitude.
10. Explain why the simple relation a′ = a is valid between Galilean inertial frames but not between frames whose origins accelerate relative to one another.

15 Answer check

1. 16 m/s east.
2. v_A∕B = +8 m/s east; v_B∕A = −8 m/s, or 8 m/s west.
3. 37 m/s east relative to B.
4. |v| = = 5.39 m/s; ϕ = tan ⁻¹(2∕5) = 21.8^∘ east of north.
5. t = 30.0 s; drift = 60.0 m east.
6. 𝜃 = sin ⁻¹(60∕250) = 13.9^∘ west of north; ground speed = = 242.7 km/h north.
7. v_P∕G = 11e_x + 4e_y m/s.
8. The same acceleration: (2e_x − 3e_y) m/s².
9. v_A∕B = 5e_x − 3e_y m/s; magnitude = 5.83 m/s.
10. Constant relative frame velocity has zero relative acceleration, so differentiating v′ = v−V gives a′ = a; an accelerating origin contributes a nonzero subtraction term.

16 Summary

Relative motion is vector bookkeeping tied to clearly identified observers. For translating frames with parallel axes,

(60)

(61)

and

(62)

For two objects measured in the same frame,

(63)

When the relative frame velocity is constant, the acceleration is the same in both Galilean inertial frames.

References